Functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. A parabola is a specific type of function.
Relative to
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
These properties concern the
domain, 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 th ...
and the
image of functions.
*
Injective function: has a distinct value for each distinct argument. Also called an injection or, sometimes, one-to-one function. In other words, every element of the function's codomain is the image of at most one element of its domain.
*
Surjective function
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 ...
: has a
preimage for every element of 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 th ...
, that is, the codomain equals the image. Also called a surjection or
onto function.
*
Bijective function
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 ...
: is both an
injection
Injection or injected may refer to:
Science and technology
* Injective function, a mathematical function mapping distinct arguments to distinct values
* Injection (medicine), insertion of liquid into the body with a syringe
* Injection, in broadca ...
and a
surjection
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 ...
, and thus
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
.
*
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 ...
: maps any given element to itself.
*
Constant function: has a fixed value regardless of arguments.
*
Empty function
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
: whose domain equals the
empty set.
*
Set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R an ...
: whose input is a set.
*
Choice function called also
selector
Selector may refer to:
*Selector, electrical or mechanical component, a switch
*''Selector'', music scheduling software for radio stations created by Radio Computing Services
*Selector, of music, otherwise known as a disc jockey
*Selector, a per ...
or
uniformizing function: assigns to each set one of its elements.
Relative to an operator (c.q. a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
or other
structure)
These properties concern how the function is affected by
arithmetic operations on its operand.
The following are special examples of a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
on a
binary operation:
*
Additive function
In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer variable ''n'' such that whenever ''a'' and ''b'' are coprime, the function applied to the product ''ab'' is the sum of the values of the func ...
: preserves the addition operation: ''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'').
*
Multiplicative function
In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and
f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime.
An arithmetic function ''f''(''n'') i ...
: preserves the multiplication operation: ''f''(''xy'') = ''f''(''x'')''f''(''y'').
Relative to
negation:
*
Even function: is symmetric with respect to the ''Y''-axis. Formally, for each ''x'': ''f''(''x'') = ''f''(−''x'').
*
Odd function
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power se ...
: is symmetric with respect to the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
. Formally, for each ''x'': ''f''(−''x'') = −''f''(''x'').
Relative to a binary operation and an
order:
*
Subadditive function In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element ...
: for which the value of ''f''(''x''+''y'') is less than or equal to ''f''(''x'') + ''f''(''y'').
*
Superadditive function: for which the value of ''f''(''x''+''y'') is greater than or equal to ''f''(''x'') + ''f''(''y'').
Relative to a topology
*
Continuous function: in which
preimages of
open set
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 su ...
s are open.
*
Nowhere continuous function: is not continuous at any point of its domain; for example, the
Dirichlet function
In mathematics, the Dirichlet function is the indicator function 1Q or \mathbf_\Q of the set of rational numbers Q, i.e. if ''x'' is a rational number and if ''x'' is not a rational number (i.e. an irrational number).
\mathbf 1_\Q(x) = \begin
1 & ...
.
*
Homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
: is a
bijective function
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 ...
that is also
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
, whose
inverse is continuous.
*
Open function: maps open sets to open sets.
*
Closed function: maps closed sets to closed sets.
*
Compactly supported function: vanishes outside a compact set.
*
Càdlàg In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset ...
function, called also RCLL function, corlol function, etc.: right-continuous, with left limits.
*
Quasi-continuous function: roughly, close to ''f''(''x'') for some but not all ''y'' near ''x'' (rather technical).
Relative to topology and order:
*
Semicontinuous function: upper or lower semicontinuous.
*
Right-continuous function: no jump when the limit point is approached from the right. Left-continuous function: similarly.
*
Locally bounded
In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number.
Locally bounded ...
function: bounded around every point.
Relative to an ordering
*
Monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
: does not reverse ordering of any pair.
* Strict
Monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
: preserves the given order.
Relative to the real/complex/hypercomplex/P-adic numbers
*
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 ...
: a function whose domain is
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) ...
.
*
Complex function: a function whose domain is
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 ...
.
*
Holomorphic function:
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 ...
valued function of a complex variable which is differentiable at every point in its domain.
*
Meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
:
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 ...
valued function that is holomorphic everywhere, apart from at isolated points where there are
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
.
*
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
: A
holomorphic function whose domain is the entire
complex plane.
*
Quaternionic function: a function whose domain is
quaternionic
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
.
*
Hypercomplex function: a function whose domain is
hypercomplex (e.g.
quaternions,
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s, trigintaduonions etc.)
*
P-adic function: a function whose domain is
P-adic
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
.
*
Linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
; also
affine function.
*
Convex function: line segment between any two points on the graph lies above the graph. Also
concave function.
*
Arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
: A function from the positive
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
into the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s.
*
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
: Can be defined locally by a
convergent power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
.
*
Quasi-analytic function: not analytic, but still, locally determined by its derivatives at a point.
*
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 it ...
: Has a
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. ...
.
*
Continuously 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 ...
: differentiable, with continuous derivative.
*
Smooth function: Has derivatives of all orders.
*
Lipschitz function
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 exi ...
,
Holder function: somewhat more than
uniformly continuous function
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 ...
.
*
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
: its value at the center of a ball is equal to the average value on the surface of the ball
(mean value property). Also
subharmonic function
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.
Intuitively, subharmonic functions are related to convex functio ...
and
superharmonic function.
*
Elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
: composition of arithmetic operations, exponentials, logarithms, constants, and solutions of algebraic equations.
*
Special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined b ...
: non-elementary functions that have established names and notations due to their importance.
*
Trigonometric functions
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 al ...
: relate the angles of a triangle to the lengths of its sides.
*
Nowhere differentiable function
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
The Weierstras ...
called also
Weierstrass function
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
The Weierstr ...
: continuous everywhere but not differentiable even at a single point.
*
Fast-growing (or rapidly increasing) function; in particular,
Ackermann function.
*
Simple function: a real-valued function over a subset of the real line, similar to a step function.
Relative to measurability
*
Measurable function: the preimage of each measurable set is measurable.
*
Borel function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in d ...
: the preimage of each
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
is a Borel set.
*
Baire function called also
Baire measurable function: obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions.
*
Singular function
In mathematics, a real-valued function ''f'' on the interval 'a'', ''b''is said to be singular if it has the following properties:
*''f'' is continuous on 'a'', ''b'' (**)
*there exists a set ''N'' of measure 0 such that for all ''x'' outside ...
: continuous, with zero derivative
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
, but non-constant.
Relative to measure
*
Integrable function
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
: has an integral (finite).
*
Square-integrable function: the square of its absolute value is integrable.
Relative to measure and topology
*
Locally integrable function In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies ...
: integrable around every point.
Ways of defining functions/relation to type theory
*
Polynomial function: defined by evaluating a polynomial.
*
Rational function: ratio of two polynomial functions. In particular,
Möbius transformation called also
linear fractional function.
*
Algebraic function In mathematics, an algebraic function is a function that can be defined
as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
: defined as the root of a polynomial equation.
*
Transcendental function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.
In other words, a transcendental function "transcends" algebra in that it cannot be expressed alge ...
: analytic but not algebraic. Also
hypertranscendental function.
*
Composite function: is formed by the composition of two functions ''f'' and ''g'', by mapping ''x'' to ''f''(''g''(''x'')).
*
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 ...
: is declared by "doing the reverse" of a given function (e.g.
arcsine
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
is the inverse of
sine).
*
Implicit function
In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0.
An implicit func ...
: defined implicitly by a relation between the argument(s) and the value.
*
Piecewise function: is defined by different expressions at different intervals.
*
Computable function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do ...
: an algorithm can do the job of the function. Also
semicomputable function;
primitive recursive function
In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined ...
;
partial recursive function
In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as i ...
.
In general, functions are often defined by specifying the name of a dependent variable, and a way of calculating what it should map to. For this purpose, the
symbol or
Church
Church may refer to:
Religion
* Church (building), a building for Christian religious activities
* Church (congregation), a local congregation of a Christian denomination
* Church service, a formalized period of Christian communal worship
* C ...
's
is often used. Also, sometimes mathematicians notate a function's
domain and
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 th ...
by writing e.g.
. These notions extend directly to
lambda calculus and
type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a fou ...
, respectively.
Higher order functions
These are functions that operate on functions or produce other functions, see
Higher order function.
Examples are:
*
Function composition.
*
Integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
and
Differential operations.
*
Fourier transforms.
*
Fold and
Map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
operations.
*
Currying
Relation to category theory
Category theory is a branch of mathematics that formalizes the notion of a special function via arrows or
morphisms
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
. A
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
is an algebraic object that (abstractly) consists of a class of ''objects'', and for every pair of objects, a set of ''morphisms''. A partial (equiv.
dependently typed) binary operation called
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
is provided on morphisms, every object has one special morphism from it to itself called the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
on that object, and composition and identities are required to obey certain relations.
In a so-called
concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of t ...
, the objects are associated with mathematical structures like
sets,
magmas
Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural sa ...
,
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
,
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
,
topological spaces
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 point ...
,
vector spaces
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 ...
,
metric spaces
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 ...
,
partial orders,
differentiable manifolds,
uniform spaces
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 unifor ...
, etc., and morphisms between two objects are associated with ''structure-preserving functions'' between them. In the examples above, these would be
functions, magma
homomorphisms,
group homomorphisms
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
...
, ring homomorphisms,
continuous functions
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
,
linear transformations
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 ...
(or
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
),
metric map In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous).
These maps are the morphisms in the category of metric spaces, Met (Isbell 1 ...
s,
monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
s,
differentiable
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 ...
functions, and
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 ...
functions, respectively.
As an algebraic theory, one of the advantages of category theory is to enable one to prove many general results with a minimum of assumptions. Many common notions from mathematics (e.g.
surjective,
injective,
free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between eleme ...
,
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
, finite
representation,
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
) are definable purely in category theoretic terms (cf.
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphism ...
,
epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
).
Category theory has been suggested as a foundation for mathematics on par with
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
and
type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a fou ...
(cf.
topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
).
Allegory theory[Peter Freyd, Andre Scedrov (1990). Categories, Allegories. Mathematical Library Vol 39. North-Holland. .] provides a generalization comparable to category theory for
relations instead of functions.
Other functions
*
Symmetric function
In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...
: value is independent of the order of its arguments
More general objects still called functions
*
Generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
: a wide generalization of Dirac delta function, able to describe
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines ...
etc.
**
Dirac delta function: useful to describe physical phenomena such as point charges.
*
Multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
: one-to-many relation.
*
Random function:
Random element In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by who commented that the “development of probability theory and expansi ...
of a set of functions.
Relative to dimension of domain and codomain
*
Scalar-valued function
*
Multivariate function
In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function ...
*
Vector-valued function
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
See also
*
List of functions
*
List of types of sets
References
{{DEFAULTSORT:Functions
Calculus
Mathematics-related lists
Number theory
Category theory
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