In
mathematics, a continuous function is a
function 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 dialect ...
induces a continuous variation of the
value 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 cognitio ...
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 mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
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 and
complex 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 words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
.
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. I ...
. 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 compute ...
, a related concept of continuity is
Scott continuity.
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 li ...
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. H ...
defined continuity of
as follows: an infinitely small increment
of the independent variable ''x'' always produces an infinitely small change
of the dependent variable ''y'' (see e.g. ''
Cours d'Analyse'', 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). 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. I ...
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 mathematic ...
denied continuity of a function at a point ''c'' unless it was defined at and on both sides of ''c'', but
Édouard Goursat 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 ...
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 Le ...
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 inter ...
, that is a
function from
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 i ...
; 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, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
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. A function with variable is ''continuous at'' the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
, if the limit of
as tends to , is equal to
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 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
(the whole
real line
In elementary mathematics, a number line is a picture of a graduated straight 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 number to a po ...
) is often called simply a continuous function; one says also that such a function is ''continuous everywhere''. For example, all
polynomial functions 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
is continuous on its whole domain, which is the closed interval
Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some
isolated points. Examples are the functions
and
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 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
and
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
be a function defined on a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...
of the set
of real numbers.
This subset
is the domain of . Some possible choices include
*
: i.e.,
is the whole set of real numbers), or, for and real numbers,
*
:
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. Othe ...
, or
*
:
is an
open interval.
In case of the domain
being defined as an open interval,
and
do not belong to
, and the values of
and
do not matter for continuity on
.
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
as ''x'' approaches ''c'' through the domain of ''f'', exists and is equal to
In mathematical notation, this is written as
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
(Here, we have assumed that the domain of ''f'' does not have any
isolated points.)
Definition in terms of neighborhoods
A
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
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
there is a neighborhood
in its domain such that
whenever
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 ...
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 po ...
s and is thus the most general definition. It follows from this definition that a function ''f'' is automatically continuous at every
isolated point 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
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of points in the domain which
converges to ''c'', the corresponding sequence
converges to
In mathematical notation,
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
as above and an element
of the domain
,
is said to be continuous at the point
when the following holds: For any positive real number
however small, there exists some positive real number
such that for all
in the domain of
with
the value of
satisfies
Alternatively written, continuity of
at
means that for every
there exists a
such that for all
:
More intuitively, we can say that if we want to get all the
values to stay in some small
neighborhood around
we simply need to choose a small enough neighborhood for the
values around
If we can do that no matter how small the
neighborhood is, then
is continuous at
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 setti ...
.
Weierstrass had required that the interval
be entirely within the domain
, 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