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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a continuous function is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
such that a continuous variation (that is a change without jump) of the
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
induces a continuous variation of the
value Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...
of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on
intuitive Intuition is the ability to acquire knowledge without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognition; ...
notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
and
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, where arguments and values of functions are
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
and
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. A stronger form of continuity is
uniform continuity In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
. In
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, especially in
domain theory Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer ...
, a related concept of continuity is
Scott continuity In mathematics, given two partially ordered sets ''P'' and ''Q'', a function ''f'': ''P'' → ''Q'' between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subs ...
. As an example, the function denoting the height of a growing flower at time would be considered continuous. In contrast, the function denoting the amount of money in a bank account at time would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.


History

A form of the epsilon–delta definition of continuity was first given by
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liber ...
in 1817.
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
defined continuity of y = f(x) as follows: an infinitely small increment \alpha of the independent variable ''x'' always produces an infinitely small change f(x+\alpha)-f(x) of the dependent variable ''y'' (see e.g. ''
Cours d'Analyse ''Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique'' is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821. The article follows the translation by Bradley and Sandifer in ...
'', p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see
microcontinuity In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or ''S''-continuity) of an internal function ''f'' at a point ''a'' is defined as follows: :for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is in ...
). The formal definition and the distinction between pointwise continuity and
uniform continuity In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
were first given by Bolzano in the 1830s but the work wasn't published until the 1930s. Like Bolzano,
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
denied continuity of a function at a point ''c'' unless it was defined at and on both sides of ''c'', but
Édouard Goursat Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his ''Cours d'analyse mathématique'', which appeared in the first decade of the twentieth century. It se ...
allowed the function to be defined only at and on one side of ''c'', and
Camille Jordan Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at ...
allowed it even if the function was defined only at ''c''. All three of those nonequivalent definitions of pointwise continuity are still in use.
Eduard Heine Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician. Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legen ...
provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
in 1854.


Real functions


Definition

A
real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interv ...
, that is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
from
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s to real numbers, can be represented by a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
in the
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
; such a function is continuous if, roughly speaking, the graph is a single unbroken
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
is the entire real line. A more mathematically rigorous definition is given below. Continuity of real functions is usually defined in terms of
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
. A function with variable is ''continuous at'' the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, if the limit of f(x), as tends to , is equal to f(c). There are several different definitions of (global) continuity of a function, which depend on the nature of its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
. A function is continuous on an
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
if the interval is contained in the domain of the function, and the function is continuous at every point of the interval. A function that is continuous on the interval (-\infty, +\infty) (the whole
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
) is often called simply a continuous function; one says also that such a function is ''continuous everywhere''. For example, all
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s are continuous everywhere. A function is continuous on a semi-open or a closed interval, if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function f(x) = \sqrt is continuous on its whole domain, which is the closed interval ,+\infty). Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...
s. Examples are the functions x \mapsto \frac and x\mapsto \tan x. When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested with their behavior near the exceptional points, one says that they are discontinuous. A partial function is ''discontinuous'' at a point, if the point belongs to the
topological closure In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of ...
of its domain, and either the point does not belong to the domain of the function, or the function is not continuous at the point. For example, the functions x\mapsto \frac and x\mapsto \sin(\frac ) are discontinuous at , and remain discontinuous whichever value is chosen for defining them at . A point where a function is discontinuous is called a ''discontinuity''. Using mathematical notation, there are several ways to define continuous functions in each of the three senses mentioned above. Let f : D \to \R be a function defined on a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
D of the set \R of real numbers. This subset D is the domain of . Some possible choices include *D = \R : i.e., D is the whole set of real numbers), or, for and real numbers, *D = , b= \ : D is a
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
, or *D = (a, b) = \ : D is an
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
. In case of the domain D being defined as an open interval, a and b do not belong to D, and the values of f(a) and f(b) do not matter for continuity on D.


Definition in terms of limits of functions

The function is ''continuous at some point'' of its domain if the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of f(x), as ''x'' approaches ''c'' through the domain of ''f'', exists and is equal to f(c). In mathematical notation, this is written as \lim_ = f(c). In detail this means three conditions: first, has to be defined at (guaranteed by the requirement that is in the domain of ). Second, the limit of that equation has to exist. Third, the value of this limit must equal f(c). (Here, we have assumed that the domain of ''f'' does not have any
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...
s.)


Definition in terms of neighborhoods

A neighborhood (mathematics), neighborhood of a point ''c'' is a set that contains, at least, all points within some fixed distance of ''c''. Intuitively, a function is continuous at a point ''c'' if the range of ''f'' over the neighborhood of ''c'' shrinks to a single point f(c) as the width of the neighborhood around ''c'' shrinks to zero. More precisely, a function ''f'' is continuous at a point ''c'' of its domain if, for any neighborhood N_1(f(c)) there is a neighborhood N_2(c) in its domain such that f(x) \in N_1(f(c)) whenever x\in N_2(c). This definition only requires that the domain and the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
are
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s and is thus the most general definition. It follows from this definition that a function ''f'' is automatically continuous at every
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...
of its domain. As a specific example, every real valued function on the set of integers is continuous.


Definition in terms of limits of sequences

One can instead require that for any sequence (mathematics), sequence (x_n)_ of points in the domain which converges to ''c'', the corresponding sequence \left(f(x_n)\right)_ converges to f(c). In mathematical notation, \forall (x_n)_ \subset D:\lim_ x_n = c \Rightarrow \lim_ f(x_n) = f(c)\,.


Weierstrass and Jordan definitions (epsilon–delta) of continuous functions

Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function f : D \to \mathbb as above and an element x_0 of the domain D, f is said to be continuous at the point x_0 when the following holds: For any positive real number \varepsilon > 0, however small, there exists some positive real number \delta > 0 such that for all x in the domain of f with x_0 - \delta < x < x_0 + \delta, the value of f(x) satisfies f\left(x_0\right) - \varepsilon < f(x) < f(x_0) + \varepsilon. Alternatively written, continuity of f : D \to \mathbb at x_0 \in D means that for every \varepsilon > 0, there exists a \delta > 0 such that for all x \in D: \left, x - x_0\ < \delta ~~\text~~ , f(x) - f(x_0), < \varepsilon. More intuitively, we can say that if we want to get all the f(x) values to stay in some small
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
around f\left(x_0\right), we simply need to choose a small enough neighborhood for the x values around x_0. If we can do that no matter how small the f(x_0) neighborhood is, then f is continuous at x_0. In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the
metric topology In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. Weierstrass had required that the interval x_0 - \delta < x < x_0 + \delta be entirely within the domain D, but Jordan removed that restriction.


Definition in terms of control of the remainder

In proofs and numerical analysis we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. A function C: ,\infty) \to [0,\infty/math> is called a control function if * ''C'' is non-decreasing *\inf_ C(\delta) = 0 A function f : D \to R is ''C''-continuous at x_0 if there exists such a neighbourhood N(x_0) that , f(x) - f(x_0), \leq C\left(\left, x - x_0\\right) \text x \in D \cap N(x_0) A function is continuous in x_0 if it is ''C''-continuous for some control function ''C''. This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions \mathcal a function is if it is for some C \in \mathcal. For example, the Lipschitz continuity, Lipschitz and Hölder continuous functions of exponent below are defined by the set of control functions \mathcal_ = \ respectively \mathcal_ = \.


Definition using oscillation

Continuity can also be defined in terms of
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
: a function ''f'' is continuous at a point x_0 if and only if its oscillation at that point is zero; in symbols, \omega_f(x_0) = 0. A benefit of this definition is that it discontinuity: the oscillation gives how the function is discontinuous at a point. This definition is useful in
descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to ot ...
to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than \varepsilon (hence a G_ set) – and gives a very quick proof of one direction of the
Lebesgue integrability condition In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of GÃ ...
. The oscillation is equivalent to the \varepsilon-\delta definition by a simple re-arrangement, and by using a limit (
lim sup In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
,
lim inf In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
) to define oscillation: if (at a given point) for a given \varepsilon_0 there is no \delta that satisfies the \varepsilon-\delta definition, then the oscillation is at least \varepsilon_0, and conversely if for every \varepsilon there is a desired \delta, the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
.


Definition using the hyperreals

Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
defined continuity of a function in the following intuitive terms: an
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
change in the independent variable corresponds to an infinitesimal change of the dependent variable (see ''Cours d'analyse'', page 34).
Non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
is a way of making this mathematically rigorous. The real line is augmented by the addition of infinite and infinitesimal numbers to form the
hyperreal numbers In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
. In nonstandard analysis, continuity can be defined as follows. (see
microcontinuity In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or ''S''-continuity) of an internal function ''f'' at a point ''a'' is defined as follows: :for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is in ...
). In other words, an infinitesimal increment of the independent variable always produces to an infinitesimal change of the dependent variable, giving a modern expression to
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
's definition of continuity.


Construction of continuous functions

Checking the continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Given f, g \colon D \to \R, then the s = f + g (defined by s(x) = f(x) + g(x) for all x\in D) is continuous in D. The same holds for the , p = f \cdot g (defined by p(x) = f(x) \cdot g(x) for all x \in D) is continuous in D. Combining the above preservations of continuity and the continuity of
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properties ...
s and of the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
I(x) = x one arrives at the continuity of all
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s such as f(x) = x^3 + x^2 - 5 x + 3 (pictured on the right). In the same way it can be shown that the r = 1/f (defined by r(x) = 1/f(x) for all x \in D such that f(x) \neq 0) is continuous in D\setminus \. This implies that, excluding the roots of g, the q = f / g (defined by q(x) = f(x)/g(x) for all x \in D, such that g(x) \neq 0) is also continuous on D\setminus \. For example, the function (pictured) y(x) = \frac is defined for all real numbers x \neq -2 and is continuous at every such point. Thus it is a continuous function. The question of continuity at x = -2 does not arise, since x = -2 is not in the domain of y. There is no continuous function F : \R \to \R that agrees with y(x) for all x \neq -2. Since the function
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
is continuous on all reals, the
sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the u ...
G(x) = \sin(x)/x, is defined and continuous for all real x \neq 0. However, unlike the previous example, ''G'' be extended to a continuous function on real numbers, by the value G(0) to be 1, which is the limit of G(x), when ''x'' approaches 0, i.e., G(0) = \lim_ \frac = 1. Thus, by setting : G(x) = \begin \frac x & \textx \ne 0\\ 1 & \textx = 0, \end the sinc-function becomes a continuous function on all real numbers. The term is used in such cases, when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points. A more involved construction of continuous functions is the
function composition In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
. Given two continuous functions g : D_g \subseteq \R \to R_g \subseteq \R \quad \text \quad f : D_f \subseteq \R \to R_f \subseteq D_g, their composition, denoted as c = g \circ f : D_f \to \R, and defined by c(x) = g(f(x)), is continuous. This construction allows stating, for example, that e^ is continuous for all x > 0.


Examples of discontinuous functions

An example of a discontinuous function is the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
H, defined by H(x) = \begin 1 & \text x \ge 0\\ 0 & \text x < 0 \end Pick for instance \varepsilon = 1/2. Then there is no around x = 0, i.e. no open interval (-\delta,\;\delta) with \delta > 0, that will force all the H(x) values to be within the of H(0), i.e. within (1/2,\;3/2). Intuitively we can think of this type of discontinuity as a sudden
jump Jumping is a form of locomotion or movement in which an organism or non-living (e.g., robotic) mechanical system propels itself through the air along a ballistic trajectory. Jump or Jumping also may refer to: Places * Jump, Kentucky or Jump S ...
in function values. Similarly, the signum or sign function \sgn(x) = \begin \;\;\ 1 & \textx > 0\\ \;\;\ 0 & \textx = 0\\ -1 & \textx < 0 \end is discontinuous at x = 0 but continuous everywhere else. Yet another example: the function f(x) = \begin \sin\left(x^\right)&\textx \neq 0\\ 0&\textx = 0 \end is continuous everywhere apart from x = 0. Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined
pathological Pathology is the study of the causal, causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when us ...
, for example,
Thomae's function Thomae's function is a real-valued function of a real variable that can be defined as: f(x) = \begin \frac &\textx = \tfrac\quad (x \text p \in \mathbb Z \text q \in \mathbb N \text\\ 0 &\textx \text \end It is named after Carl Jo ...
, f(x)=\begin 1 &\text x=0\\ \frac&\text x = \frac \text\\ 0&\textx\text. \end is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, Dirichlet's function, the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
for the set of rational numbers, D(x)=\begin 0&\textx\text (\in \R \setminus \Q)\\ 1&\textx\text (\in \Q) \end is nowhere continuous.


Properties


A useful lemma

Let f(x) be a function that is continuous at a point x_0, and y_0 be a value such f\left(x_0\right)\neq y_0. Then f(x)\neq y_0 throughout some neighbourhood of x_0. ''Proof:'' By the definition of continuity, take \varepsilon =\frac>0 , then there exists \delta>0 such that \left, f(x)-f(x_0)\ < \frac \quad \text \quad , x-x_0, < \delta Suppose there is a point in the neighbourhood , x-x_0, <\delta for which f(x)=y_0; then we have the contradiction \left, f(x_0)-y_0\ < \frac.


Intermediate value theorem

The
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two import ...
is an
existence theorem In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential ...
, based on the real number property of completeness, and states: :If the real-valued function ''f'' is continuous on the
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
, b and ''k'' is some number between f(a) and f(b), then there is some number c \in , b such that f(c) = k. For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m. As a consequence, if ''f'' is continuous on , b/math> and f(a) and f(b) differ in
sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
, then, at some point c \in , b f(c) must equal
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
.


Extreme value theorem

The
extreme value theorem In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> suc ...
states that if a function ''f'' is defined on a closed interval , b/math> (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists c \in , b/math> with f(c) \geq f(x) for all x \in , b The same is true of the minimum of ''f''. These statements are not, in general, true if the function is defined on an open interval (a, b) (or any set that is not both closed and bounded), as, for example, the continuous function f(x) = \frac, defined on the open interval (0,1), does not attain a maximum, being unbounded above.


Relation to differentiability and integrability

Every
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
f : (a, b) \to \R is continuous, as can be shown. The
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...
does not hold: for example, the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
function :f(x)=, x, = \begin \;\;\ x & \textx \geq 0\\ -x & \textx < 0 \end is everywhere continuous. However, it is not differentiable at x = 0 (but is so everywhere else). Weierstrass's function is also everywhere continuous but nowhere differentiable. The
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
''f′''(''x'') of a differentiable function ''f''(''x'') need not be continuous. If ''f′''(''x'') is continuous, ''f''(''x'') is said to be ''continuously differentiable''. The set of such functions is denoted C^1((a, b)). More generally, the set of functions f : \Omega \to \R (from an open interval (or
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of \R) \Omega to the reals) such that ''f'' is n times differentiable and such that the n-th derivative of ''f'' is continuous is denoted C^n(\Omega). See
differentiability class In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
. In the field of computer graphics, properties related (but not identical) to C^0, C^1, C^2 are sometimes called G^0 (continuity of position), G^1 (continuity of tangency), and G^2 (continuity of curvature); see Smoothness of curves and surfaces. Every continuous function f : , b\to \R is
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
(for example in the sense of the
Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
). The converse does not hold, as the (integrable, but discontinuous)
sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avoi ...
shows.


Pointwise and uniform limits

Given a sequence (mathematics), sequence f_1, f_2, \dotsc : I \to \R of functions such that the limit f(x) := \lim_ f_n(x) exists for all x \in D,, the resulting function f(x) is referred to as the pointwise limit of the sequence of functions \left(f_n\right)_. The pointwise limit function need not be continuous, even if all functions f_n are continuous, as the animation at the right shows. However, ''f'' is continuous if all functions f_n are continuous and the sequence
converges uniformly In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
, by the uniform convergence theorem. This theorem can be used to show that the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
s,
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s,
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
function, and
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s are continuous.


Directional and semi-continuity

Image:Right-continuous.svg, A right-continuous function Image:Left-continuous.svg, A left-continuous function
Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and
semi-continuity In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, rou ...
. Roughly speaking, a function is if no jump occurs when the limit point is approached from the right. Formally, ''f'' is said to be right-continuous at the point ''c'' if the following holds: For any number \varepsilon > 0 however small, there exists some number \delta > 0 such that for all ''x'' in the domain with c < x < c + \delta, the value of f(x) will satisfy , f(x) - f(c), < \varepsilon. This is the same condition as for continuous functions, except that it is required to hold for ''x'' strictly larger than ''c'' only. Requiring it instead for all ''x'' with c - \delta < x < c yields the notion of functions. A function is continuous if and only if it is both right-continuous and left-continuous. A function ''f'' is if, roughly, any jumps that might occur only go down, but not up. That is, for any \varepsilon > 0, there exists some number \delta > 0 such that for all ''x'' in the domain with , x - c, < \delta, the value of f(x) satisfies f(x) \geq f(c) - \epsilon. The reverse condition is .


Continuous functions between metric spaces

The concept of continuous real-valued functions can be generalized to functions between
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s. A metric space is a set X equipped with a function (called
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
) d_X, that can be thought of as a measurement of the distance of any two elements in ''X''. Formally, the metric is a function d_X : X \times X \to \R that satisfies a number of requirements, notably the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
. Given two metric spaces \left(X, d_X\right) and \left(Y, d_Y\right) and a function f : X \to Y then f is continuous at the point c \in X (with respect to the given metrics) if for any positive real number \varepsilon > 0, there exists a positive real number \delta > 0 such that all x \in X satisfying d_X(x, c) < \delta will also satisfy d_Y(f(x), f(c)) < \varepsilon. As in the case of real functions above, this is equivalent to the condition that for every sequence \left(x_n\right) in X with limit \lim x_n = c, we have \lim f\left(x_n\right) = f(c). The latter condition can be weakened as follows: f is continuous at the point c if and only if for every convergent sequence \left(x_n\right) in X with limit c, the sequence \left(f\left(x_n\right)\right) is a
Cauchy sequence In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
, and c is in the domain of f. The set of points at which a function between metric spaces is continuous is a G_ set â€“ this follows from the \varepsilon-\delta definition of continuity. This notion of continuity is applied, for example, in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. A key statement in this area says that a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
T : V \to W between
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
s V and W (which are
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s equipped with a compatible
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
, denoted \, x\, ) is continuous if and only if it is bounded, that is, there is a constant K such that \, T(x)\, \leq K \, x\, for all x \in V.


Uniform, Hölder and Lipschitz continuity

The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way \delta depends on \varepsilon and ''c'' in the definition above. Intuitively, a function ''f'' as above is
uniformly continuous In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
if the \delta does not depend on the point ''c''. More precisely, it is required that for every
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
\varepsilon > 0 there exists \delta > 0 such that for every c, b \in X with d_X(b, c) < \delta, we have that d_Y(f(b), f(c)) < \varepsilon. Thus, any uniformly continuous function is continuous. The converse does not hold in general, but holds when the domain space ''X'' is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
. Uniformly continuous maps can be defined in the more general situation of
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s. A function is
Hölder continuous Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...
with exponent α (a real number) if there is a constant ''K'' such that for all b, c \in X, the inequality d_Y (f(b), f(c)) \leq K \cdot (d_X (b, c))^\alpha holds. Any Hölder continuous function is uniformly continuous. The particular case \alpha = 1 is referred to as
Lipschitz continuity In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
. That is, a function is Lipschitz continuous if there is a constant ''K'' such that the inequality d_Y (f(b), f(c)) \leq K \cdot d_X (b, c) holds for any b, c \in X. The Lipschitz condition occurs, for example, in the
Picard–Lindelöf theorem In mathematics â€“ specifically, in differential equations â€“ the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauc ...
concerning the solutions of
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s.


Continuous functions between topological spaces

Another, more abstract, notion of continuity is continuity of functions between
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s in which there generally is no formal notion of distance, as there is in the case of
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s. A topological space is a set ''X'' together with a topology on ''X'', which is a set of
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of ''X'' satisfying a few requirements with respect to their unions and intersections that generalize the properties of the
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
s in metric spaces while still allowing to talk about the
neighbourhoods A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of a given point. The elements of a topology are called
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
s of ''X'' (with respect to the topology). A function f : X \to Y between two topological spaces ''X'' and ''Y'' is continuous if for every open set V \subseteq Y, the
inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
f^(V) = \ is an open subset of ''X''. That is, ''f'' is a function between the sets ''X'' and ''Y'' (not on the elements of the topology T_X), but the continuity of ''f'' depends on the topologies used on ''X'' and ''Y''. This is equivalent to the condition that the preimages of the
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
s (which are the complements of the open subsets) in ''Y'' are closed in ''X''. An extreme example: if a set ''X'' is given the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
(in which every subset is open), all functions f : X \to T to any topological space ''T'' are continuous. On the other hand, if ''X'' is equipped with the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
(in which the only open subsets are the empty set and ''X'') and the space ''T'' set is at least T0, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.


Continuity at a point

The translation in the language of neighborhoods of the (\varepsilon, \delta)-definition of continuity leads to the following definition of the continuity at a point: This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
s rather than images. Also, as every set that contains a neighborhood is also a neighborhood, and f^(V) is the largest subset of such that f(U) \subseteq V, this definition may be simplified into: As an open set is a set that is a neighborhood of all its points, a function f : X \to Y is continuous at every point of if and only if it is a continuous function. If ''X'' and ''Y'' are metric spaces, it is equivalent to consider the
neighborhood system In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
of
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
s centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This gives back the above \varepsilon-\delta definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If however the target space is a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
, it is still true that ''f'' is continuous at ''a'' if and only if the limit of ''f'' as ''x'' approaches ''a'' is ''f''(''a''). At an isolated point, every function is continuous. Given x \in X, a map f : X \to Y is continuous at x if and only if whenever \mathcal is a filter on X that converges to x in X, which is expressed by writing \mathcal \to x, then necessarily f(\mathcal) \to f(x) in Y. If \mathcal(x) denotes the
neighborhood filter In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbou ...
at x then f : X \to Y is continuous at x if and only if f(\mathcal(x)) \to f(x) in Y. Moreover, this happens if and only if the
prefilter In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then ...
f(\mathcal(x)) is a
filter base In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then ...
for the neighborhood filter of f(x) in Y.


Alternative definitions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.


Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of
limit points In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
. In many instances, this is accomplished by specifying when a point is the
limit of a sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...
, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
, known as nets. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition. In detail, a function f : X \to Y is
sequentially continuous In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
if whenever a sequence \left(x_n\right) in X converges to a limit x, the sequence \left(f\left(x_n\right)\right) converges to f(x). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If X is a
first-countable space In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called
sequential space In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
s.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions. For instance, consider the case of real-valued functions of one real variable: ''Proof.'' Assume that f : A \subseteq \R \to \R is continuous at x_0 (in the sense of \epsilon-\delta continuity). Let \left(x_n\right)_ be a sequence converging at x_0 (such a sequence always exists, for example, x_n = x, \text n); since f is continuous at x_0 \forall \epsilon > 0\, \exists \delta_ > 0 : 0 < , x-x_0, < \delta_ \implies , f(x)-f(x_0), < \epsilon.\quad (*) For any such \delta_ we can find a natural number \nu_ > 0 such that for all n > \nu_, , x_n-x_0, < \delta_, since \left(x_n\right) converges at x_0; combining this with (*) we obtain \forall \epsilon > 0 \,\exists \nu_ > 0 : \forall n > \nu_ \quad , f(x_n)-f(x_0), < \epsilon. Assume on the contrary that f is sequentially continuous and proceed by contradiction: suppose f is not continuous at x_0 \exists \epsilon > 0 : \forall \delta_ > 0,\,\exists x_: 0 < , x_-x_0, < \delta_\epsilon \implies , f(x_)-f(x_0), > \epsilon then we can take \delta_=1/n,\,\forall n > 0 and call the corresponding point x_ =: x_n: in this way we have defined a sequence (x_n)_ such that \forall n > 0 \quad , x_n-x_0, < \frac,\quad , f(x_n)-f(x_0), > \epsilon by construction x_n \to x_0 but f(x_n) \not\to f(x_0), which contradicts the hypothesis of sequentially continuity. \blacksquare


Closure operator and interior operator definitions

In terms of the interior operator, a function f : X \to Y between topological spaces is continuous if and only if for every subset B \subseteq Y, f^\left(\operatorname_Y B\right) ~\subseteq~ \operatorname_X\left(f^(B)\right). In terms of the closure operator, f : X \to Y is continuous if and only if for every subset A \subseteq X, f\left(\operatorname_X A\right) ~\subseteq~ \operatorname_Y (f(A)). That is to say, given any element x \in X that belongs to the closure of a subset A \subseteq X, f(x) necessarily belongs to the closure of f(A) in Y. If we declare that a point x is a subset A \subseteq X if x \in \operatorname_X A, then this terminology allows for a
plain English Plain English (or layman's terms) are groups of words that are to be clear and easy to know. It usually avoids the use of rare words and uncommon euphemisms to explain the subject. Plain English wording is intended to be suitable for almost anyone, ...
description of continuity: f is continuous if and only if for every subset A \subseteq X, f maps points that are close to A to points that are close to f(A). Similarly, f is continuous at a fixed given point x \in X if and only if whenever x is close to a subset A \subseteq X, then f(x) is close to f(A). Instead of specifying topological spaces by their
open subsets In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
, any topology on X can alternatively be determined by a
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are dete ...
or by an
interior operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are det ...
. Specifically, the map that sends a subset A of a topological space X to its
topological closure In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of ...
\operatorname_X A satisfies the
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first forma ...
. Conversely, for any
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are dete ...
A \mapsto \operatorname A there exists a unique topology \tau on X (specifically, \tau := \) such that for every subset A \subseteq X, \operatorname A is equal to the topological closure \operatorname_ A of A in (X, \tau). If the sets X and Y are each associated with closure operators (both denoted by \operatorname) then a map f : X \to Y is continuous if and only if f(\operatorname A) \subseteq \operatorname (f(A)) for every subset A \subseteq X. Similarly, the map that sends a subset A of X to its topological interior \operatorname_X A defines an
interior operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are det ...
. Conversely, any interior operator A \mapsto \operatorname A induces a unique topology \tau on X (specifically, \tau := \) such that for every A \subseteq X, \operatorname A is equal to the topological interior \operatorname_ A of A in (X, \tau). If the sets X and Y are each associated with interior operators (both denoted by \operatorname) then a map f : X \to Y is continuous if and only if f^(\operatorname B) \subseteq \operatorname\left(f^(B)\right) for every subset B \subseteq Y.


Filters and prefilters

Continuity can also be characterized in terms of
filters Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component tha ...
. A function f : X \to Y is continuous if and only if whenever a filter \mathcal on X converges in X to a point x \in X, then the
prefilter In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then ...
f(\mathcal) converges in Y to f(x). This characterization remains true if the word "filter" is replaced by "prefilter."


Properties

If f : X \to Y and g : Y \to Z are continuous, then so is the composition g \circ f : X \to Z. If f : X \to Y is continuous and * ''X'' is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
, then ''f''(''X'') is compact. * ''X'' is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
, then ''f''(''X'') is connected. * ''X'' is
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
, then ''f''(''X'') is path-connected. * ''X'' is Lindelöf, then ''f''(''X'') is Lindelöf. * ''X'' is separable, then ''f''(''X'') is separable. The possible topologies on a fixed set ''X'' are
partially ordered In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
: a topology \tau_1 is said to be coarser than another topology \tau_2 (notation: \tau_1 \subseteq \tau_2) if every open subset with respect to \tau_1 is also open with respect to \tau_2. Then, the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
\operatorname_X : \left(X, \tau_2\right) \to \left(X, \tau_1\right) is continuous if and only if \tau_1 \subseteq \tau_2 (see also
comparison of topologies In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the ...
). More generally, a continuous function \left(X, \tau_X\right) \to \left(Y, \tau_Y\right) stays continuous if the topology \tau_Y is replaced by a
coarser topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as th ...
and/or \tau_X is replaced by a
finer topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the ...
.


Homeomorphisms

Symmetric to the concept of a continuous map is an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a ...
, for which of open sets are open. In fact, if an open map ''f'' has an
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\t ...
, that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open. Given a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
function ''f'' between two topological spaces, the inverse function f^ need not be continuous. A bijective continuous function with continuous inverse function is called a . If a continuous bijection has as its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
and its codomain is Hausdorff, then it is a homeomorphism.


Defining topologies via continuous functions

Given a function f : X \to S, where ''X'' is a topological space and ''S'' is a set (without a specified topology), the
final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that make ...
on ''S'' is defined by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which f^(A) is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on ''S''. Thus the final topology can be characterized as the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
, this topology is canonically identified with the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
under the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
defined by ''f''. Dually, for a function ''f'' from a set ''S'' to a topological space ''X'', the
initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' tha ...
on ''S'' is defined by designating as an open set every subset ''A'' of ''S'' such that A = f^(U) for some open subset ''U'' of ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on ''S''. Thus the initial topology can be characterized as the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
of ''S'', viewed as a subset of ''X''. A topology on a set ''S'' is uniquely determined by the class of all continuous functions S \to X into all topological spaces ''X''.
Dually Dually may refer to: *Dualla, County Tipperary, a village in Ireland *A pickup truck with dual wheels on the rear axle * DUALLy, s platform for architectural languages interoperability * Dual-processor See also * Dual (disambiguation) Dual or ...
, a similar idea can be applied to maps X \to S.


Related notions

If f : S \to Y is a continuous function from some subset S of a topological space X then a of f to X is any continuous function F : X \to Y such that F(s) = f(s) for every s \in S, which is a condition that often written as f = F\big\vert_S. In words, it is any continuous function F : X \to Y that restricts to f on S. This notion is used, for example, in the
Tietze extension theorem In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness ...
and the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
. Were f : S \to Y not continuous then it could not possibly have a continuous extension. If Y is a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
and S is a
dense subset In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of X then a continuous extension of f : S \to Y to X, if one exists, will be unique. The
Blumberg theorem In mathematics, the Blumberg theorem states that for any real function f : \R \to \R there is a Dense set, dense subset D of \mathbb such that the Restriction_(mathematics), restriction of f to D is continuous function, continuous. For instance, t ...
states that if f : \R \to \R is an arbitrary function then there exists a dense subset D of \R such that the restriction f\big\vert_D : D \to \R is continuous; in other words, every function \R \to \R can be restricted to some dense subset on which it is continuous. Various other mathematical domains use the concept of continuity in different, but related meanings. For example, in
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, an order-preserving function f : X \to Y between particular types of
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
s X and Y is continuous if for each directed subset A of X, we have \sup f(A) = f(\sup A). Here \,\sup\, is the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
with respect to the orderings in X and Y, respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the
Scott topology Scott may refer to: Places Canada * Scott, Quebec, municipality in the Nouvelle-Beauce regional municipality in Quebec * Scott, Saskatchewan, a town in the Rural Municipality of Tramping Lake No. 380 * Rural Municipality of Scott No. 98, Saskat ...
. In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
F : \mathcal C \to \mathcal D between two
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
is called if it commutes with small
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
. That is to say, \varprojlim_ F(C_i) \cong F \left(\varprojlim_ C_i \right) for any small (that is, indexed by a set I, as opposed to a
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differentl ...
)
diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three- ...
of
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
in \mathcal C. A is a generalization of metric spaces and posets, which uses the concept of
quantale In mathematics, quantales are certain partially ordered algebraic structures that generalize locales ( point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis ( C*-algebras, von Neumann ...
s, and that can be used to unify the notions of metric spaces and
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
s.


See also

*
Continuity (mathematics) In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways. Continuity of functions and measures * Continuous function * Absolutely continuous function * Absolute continuity of a measure with respec ...
*
Absolute continuity In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
*
Dini continuity In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous. Definition Let X be a compact subset of a metric space (such as \mathbb ...
*
Equicontinuity In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable fa ...
*
Geometric continuity In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if i ...
*
Parametric continuity In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
*
Classification of discontinuities Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of ...
*
Coarse function In mathematics, coarse functions are functions that may appear to be continuous at a distance, but in reality are not necessarily continuous.Chul-Woo Lee and Jared Duke (2007)Coarse Function Value Theorems ''Rose-Hulman Undergraduate Mathematics J ...
*
Continuous function (set theory) In set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits ( limit suprema and limit infima) of all values at previous stages. More formally, let γ be an ordinal, and s := \langle s_, ...
*
Continuous stochastic process In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be " continuous" as a function of its "time" or index parameter. Continuity is a nice property for (the sample paths of) a process to have ...
*
Normal function In axiomatic set theory, a function ''f'' : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two c ...
*
Open and closed maps In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a ...
*
Piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
*
Symmetrically continuous function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
* Direction-preserving function - an analogue of a continuous function in discrete spaces.


References


Bibliography

* * {{Authority control Calculus Types of functions