Function (mathematics)

In mathematics,[] a function is a binary relation over two sets that associates to every element of the first set exactly one element of the second set. Typical examples are functions from integers to integers or from the real numbers to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable. The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that associates each element of a set, the domain of the function, to a single element of another set , the codomain of the function. If the function is called, this relation is denoted , the element is the argument or input of the function, and is the value of the function, the output, or the image of by. The symbol that is used for representing the input is the variable of the function.
A function is uniquely represented by the set of all pairs, called the graph of the function. When the domain and the codomain are sets of real numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means to illustrate the function.
Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.
" that for each input yields a corresponding output
, because any vertical line has exactly one crossing point with the curve.

Definition

Intuitively, a function is a process that associates to each element of a set a single element of a set.
Formally, a function from a set to a set is defined by a set of ordered pairs such that,, and every element of is the first component of exactly one ordered pair in. In other words, for every in, there is exactly one element such that the ordered pair belongs to the set of pairs defining the function. The set is called the graph of the function. Formally speaking, it may be identified with the function, but this hides the usual interpretation of a function as a process. Therefore, in common usage, the function is generally distinguished from its graph. Functions are also called maps or mappings, though some authors make some distinction between "maps" and "functions".
In the definition of function, and are respectively called the domain and the codomain of the function. If belongs to the set defining, then is the image of under, or the value of applied to the argument. Especially in the context of numbers, one says also that is the value of for the value of its variable, or, still shorter, is the value of of, denoted as.
Two functions and are equal if their domain and codomain sets are the same and their output values agree on the whole domain. Formally, if for all, where and.
The domain and codomain are not always explicitly given when a function is defined, and, without some computation, one knows only that the domain is contained in a larger set. Typically, this occurs in mathematical analysis, where "a function often refers to a function that may have a proper subset of as domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable, and this phrase does not mean that the domain of the function is the whole set of the real numbers, but only that the domain is a set of real numbers that contains a non-empty open interval; such a function is then called a partial function. For example, if is a function that has the real numbers as domain and codomain, then a function mapping the value to the value is a function from the reals to the reals, whose domain is the set of the reals, such that.
The range of a function is the set of the images of all elements in the domain. However, range is sometimes used as a synonym of codomain, generally in old textbooks.

Relational approach

Any subset of the Cartesian product of two sets and defines a binary relation between these two sets. It is immediate that an arbitrary relation may contain pairs that violate the necessary conditions for a function, given above.
A binary relation is functional if
A binary relation is serial if
A partial function is a binary relation that is functional.
A function is a binary relation that is functional and serial.
Various properties of functions and function [|composition] may be reformulated in the language of relations. For example, a function is injective if the converse relation is functional, where the converse relation is defined as

As an element of a Cartesian product over a domain

The set of all functions from some given domain to a codomain is sometimes identified with the Cartesian product of copies of the codomain, indexed by the domain. Namely, given sets and any function is an element of the Cartesian product of copies of s over the index set
Viewing as tuple with coordinates, then for each, the th coordinate of this tuple is the value This reflects the intuition that for each the function picks some element namely,
Infinite Cartesian products are often simply "defined" as sets of functions.

Notation

There are various standard ways for denoting functions. The most commonly used notation is functional notation, which defines the function using an equation that gives the names of the function and the argument explicitly. This gives rise to a subtle point, often glossed over in elementary treatments of functions: functions are distinct from their values. Thus, a function should be distinguished from its value at the value in its domain. To some extent, even working mathematicians will conflate the two in informal settings for convenience, and to avoid appearing pedantic. However, strictly speaking, it is an abuse of notation to write "let be the function ", since and should both be understood as the value of f at x, rather than the function itself. Instead, it is correct, though long-winded, to write "let be the function defined by the equation valid for all real values of ". A compact phrasing is "let with " where the redundant "be the function" is omitted and, by convention, "for all in the domain of " is understood.
This distinction in language and notation becomes important in cases where functions themselves serve as inputs for other functions. Other approaches to denoting functions, detailed below, avoid this problem but are less commonly used.

Functional notation

As first used by Leonhard Euler in 1734, functions are denoted by a symbol consisting generally of a single letter in italic font, most often the lower-case letters. Some widely used functions are represented by a symbol consisting of several letters. By convention, in this case, a roman type is used, such as "" for the sine function, in contrast to italic font for single-letter symbols.
The notation
means that the pair belongs to the set of pairs defining the function. If is the domain of, the set of pairs defining the function is thus, using set-builder notation,
Often, a definition of the function is given by what f does to the explicit argument x. For example, a function f can be defined by the equation
for all real numbers x. In this example, f can be thought of as the [|composite] of several simpler functions: squaring, adding 1, and taking the sine. However, only the sine function has a common explicit symbol, while the combination of squaring and then adding 1 is described by the polynomial expression. In order to explicitly reference functions such as squaring or adding 1 without introducing new function names, one of the methods below could be used.
Sometimes the parentheses of functional notation are omitted when the symbol denoting the function consists of several characters and no ambiguity may arise. For example, can be written instead of

Arrow notation

For explicitly expressing domain and the codomain of a function, the arrow notation is often used :
or
This is often used in relation with the arrow notation for elements, often stacked immediately below the arrow notation giving the function symbol, domain, and codomain:
For example, if a multiplication is defined on a set, then the square function on is unambiguously defined by
the latter line being more commonly written
Often, the expression giving the function symbol, domain and codomain is omitted. Thus, the arrow notation is useful for avoiding introducing a symbol for a function that is defined, as it is often the case, by a formula expressing the value of the function in terms of its argument. As a common application of the arrow notation, suppose is a two-argument function, and we want to refer to a partially applied function produced by fixing the second argument to the value without introducing a new function name. The map in question could be denoted using the arrow notation for elements. The expression represents this new function with just one argument, whereas the expression refers to the value of the function at the

Index notation

Index notation is often used instead of functional notation. That is, instead of writing, one writes
This is typically the case for functions whose domain is the set of the natural numbers. Such a function is called a sequence, and, in this case the element is called the th element of sequence.
The index notation is also often used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map would be denoted using index notation, if we define the collection of maps by the formula for all.

Dot notation

In the notation
the symbol does not represent any value, it is simply a placeholder meaning that, if is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, may be replaced by any symbol, often an interpunct "". This may be useful for distinguishing the function from its value at.
For example, may stand for the function, and may stand for a function defined by an integral with variable upper bound:.

Specialized notations

There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis, linear forms and the vectors they act upon are denoted using a dual pair to show the underlying duality. This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application. In category theory and homological algebra, networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.

Other terms

Map

A function is often also called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure. In particular map is often used in place of homomorphism for the sake of succinctness. Some authors reserve the word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function.
Some authors, such as Serge Lang, use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term mapping for more general functions.
In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.
Whichever definition of map is used, related terms like domain, codomain, injective, continuous have the same meaning as for a function.

Specifying a function

Given a function, by definition, to each element of the domain of the function, there is a unique element associated to it, the value of at. There are several ways to specify or describe how is related to, both explicitly and implicitly. Sometimes, a theorem or an axiom asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function.

By listing function values

On a finite set, a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. E.g., if, then one can define a function by

By a formula

Functions are often defined by a formula that describes a combination of arithmetic operations and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain.
For example, in the above example, can be defined by the formula, for.
When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the zeros of auxiliary functions. Similarly, if square roots occur in the definition of a function from to the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.
For example, defines a function whose domain is because is always positive if is a real number. On the other hand, defines a function from the reals to the reals whose domain is reduced to the interval.
Functions are often classified by the nature of formulas that can that define them:
A function with domain and codomain, is bijective, if for every in, there is one and only one element in such that. In this case, the inverse function of is the function that maps to the element such that. For example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It has these an inverse, called the exponential function that maps the real numbers onto the positive numbers.
If a function is not bijective, it may occur that one can select subsets and such that the restriction of to is a bijection from to, and has thus an inverse. The inverse trigonometric functions are defined this way. For example, the cosine function induces, by restriction, a bijection from the interval onto the interval, and its inverse function, called arccosine, maps onto. The other inverse trigonometric functions are defined similarly.
More generally, given a binary relation between two sets and, let be a subset of such that, for every there is some such that. If one has a criterion allowing selecting such an for every this defines a function called an implicit function, because it is implicitly defined by the relation.
For example, the equation of the unit circle defines a relation on real numbers. If there are two possible values of, one positive and one negative. For, these two values become both equal to 0. Otherwise, there is no possible value of. This means that the equation defines two implicit functions with domain and respective codomains and.
In this example, the equation can be solved in, giving but, in more complicated examples, this is impossible. For example, the relation defines as an implicit function of, called the Bring radical, which has as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and th roots.
The implicit function theorem provides mild differentiability conditions for existence and uniqueness of an implicit function in the neighborhood of a point.

Using differential calculus

Many functions can be defined as the antiderivative of another function. This is the case of the natural logarithm, which is the antiderivative of that is 0 for. Another common example is the error function.
More generally, many functions, including most special functions, can be defined as solutions of differential equations. The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for.
Power series can be used to define functions on the domain in which they converge. For example, the exponential function is given by. However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its Taylor series in some interval, this power series allows immediately enlarging the domain to a subset of the complex numbers, the disc of convergence of the series. Then analytic continuation allows enlarging further the domain for including almost the whole complex plane. This process is the method that is generally used for defining the logarithm, the exponential and the trigonometric functions of a complex number.

By recurrence

Functions whose domain are the nonnegative integers, known as sequences, are often defined by recurrence relations.
The factorial function on the nonnegative integers is a basic example, as it can be defined by the recurrence relation
and the initial condition

Representing a function

A graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.

Graphs and plots

Given a function its graph is, formally, the set
In the frequent case where and are subsets of the real numbers, an element may be identified with a point having coordinates in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a plot that represents the function. The use of plots is so ubiquitous that they too are called the graph of the function. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the square function
consisting of all points with coordinates for yields, when depicted in Cartesian coordinates, the well known parabola. If the same quadratic function with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates the plot obtained is Fermat's spiral.

Tables

A function can be represented as a table of values. If the domain of a function is finite, then the function can be completely specified in this way. For example, the multiplication function defined as can be represented by the familiar multiplication table
 1 2 3 4 5 1 1 2 3 4 5 2 2 4 6 8 10 3 3 6 9 12 15 4 4 8 12 16 20 5 5 10 15 20 25

On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed, interpolation can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:
 1.289 0.960557 1.290 0.960835 1.291 0.961112 1.292 0.961387 1.293 0.961662

Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.

Bar chart

Bar charts are often used for representing functions whose domain is a finite set, the natural numbers, or the integers. In this case, an element of the domain is represented by an interval of the -axis, and the corresponding value of the function,, is represented by a rectangle whose base is the interval corresponding to and whose height is .

General properties

This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.

Standard functions

There are a number of standard functions that occur frequently:
• For every set, there is a unique function, called the from the empty set to. The graph of an empty function is the empty set. The existence of the empty function is a convention that is needed for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements.
• For every set and every singleton set, there is a unique function from to, which maps every element of to. This is a surjection unless is the empty set.
• Given a function the canonical surjection of onto its image is the function from to that maps to.
• For every subset of a set, the inclusion map of into is the injective function that maps every element of to itself.
• The identity function on a set, often denoted by, is the inclusion of into itself.

Function composition

Given two functions and such that the domain of is the codomain of, their composition is the function defined by
That is, the value of is obtained by first applying to to obtain and then applying to the result to obtain. In the notation the function that is applied first is always written on the right.
The composition is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both and satisfy these conditions, the composition is not necessarily commutative, that is, the functions and need not be equal, but may deliver different values for the same argument. For example, let and, then and agree just for
The function composition is associative in the sense that, if one of and is defined, then the other is also defined, and they are equal. Thus, one writes
The identity functions and are respectively a right identity and a left identity for functions from to. That is, if is a function with domain, and codomain, one has

Image and preimage

Let The image by of an element of the domain is. If is any subset of, then the image of by, denoted is the subset of the codomain consisting of all images of elements of, that is,
The image of is the image of the whole domain, that is. It is also called the range of, although the term may also refer to the codomain.
On the other hand, the inverse image, or preimage by of a subset of the codomain is the subset of the domain consisting of all elements of whose images belong to. It is denoted by That is
For example, the preimage of under the square function is the set.
By definition of a function, the image of an element of the domain is always a single element of the codomain. However, the preimage of a single element, denoted may be empty or contain any number of elements. For example, if is the function from the integers to themselves that maps every integer to 0, then.
If is a function, and are subsets of, and and are subsets of, then one has the following properties:
The preimage by of an element of the codomain is sometimes called, in some contexts, the fiber of under.
If a function has an inverse, this inverse is denoted In this case may denote either the image by or the preimage by of. This is not a problem, as these sets are equal. The notation and may be ambiguous in the case of sets that contain some subsets as elements, such as In this case, some care may be needed, for example, by using square brackets for images and preimages of subsets, and ordinary parentheses for images and preimages of elements.

Injective, surjective and bijective functions

Let be a function.
The function is injective if for any two different elements and of. Equivalently, is injective if, for any the preimage contains at most one element. An empty function is always injective. If is not the empty set, and if, as usual, Zermelo–Fraenkel set theory is assumed, then is injective if and only if there exists a function such that that is, if has a left inverse. If is injective, for defining, one chooses an element in , and one defines by if and, if
The function is surjective if the range equals the codomain, that is, if. In other words, the preimage of every is nonempty. If, as usual, the axiom of choice is assumed, then is surjective if and only if there exists a function such that that is, if has a right inverse. The axiom of choice is needed, because, if is surjective, one defines by where is an arbitrarily chosen element of
The function is bijective if it is both injective and surjective. That is is bijective if, for any the preimage contains exactly one element. The function is bijective if and only if it admits an inverse function, that is a function such that and
Every function may be factorized as the composition of a surjection followed by an injection, where is the canonical surjection of onto, and is the canonical injection of into. This is the canonical factorization of.
"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement " maps onto " differs from " maps into " in that the former implies that is surjective, while the latter makes no assertion about the nature of the mapping. In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage to be more symmetrical.

Restriction and extension

If is a function and S is a subset of X, then the restriction of to S, denoted, is the function from S to Y defined by
for all x in S. Restrictions can be used to define partial inverse functions: if there is a subset S of the domain of a function such that is injective, then the canonical surjection of onto its image is a bijection, and thus has an inverse function from to S. One application is the definition of inverse trigonometric functions. For example, the cosine function is injective when restricted to the interval. The image of this restriction is the interval, and thus the restriction has an inverse function from to, which is called arccosine and is denoted.
Function restriction may also be used for "gluing" functions together. Let be the decomposition of as a union of subsets, and suppose that a function is defined on each such that for each pair of indices, the restrictions of and to are equal. Then this defines a unique function such that for all. This is the way that functions on manifolds are defined.
An extension of a function is a function such that is a restriction of. A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.
Here is another classical example of a function extension that is encountered when studying homographies of the real line. A homography is a function such that. Its domain is the set of all real numbers different from and its image is the set of all real numbers different from If one extends the real line to the projectively extended real line by including, one may extend to a bijection from the extended real line to itself by setting and.

Multivariate function

A multivariate function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time and its speed.
More formally, a function of variables is a function whose domain is a set of -tuples.
For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all pairs of integers, and whose codomain is the set of integers. The same is true for every binary operation. More generally, every mathematical operation is defined as a multivariate function.
The Cartesian product of sets is the set of all -tuples such that for every with. Therefore, a function of variables is a function
where the domain has the form
When using function notation, one usually omits the parentheses surrounding tuples, writing instead of
In the case where all the are equal to the set of real numbers, one has a function of several real variables. If the are equal to the set of complex numbers, one has a function of several complex variables.
It is common to also consider functions whose codomain is a product of sets. For example, Euclidean division maps every pair of integers with to a pair of integers called the quotient and the remainder:
The codomain may also be a vector space. In this case, one talks of a vector-valued function. If the domain is contained in a Euclidean space, or more generally a manifold, a vector-valued function is often called a vector field.

In calculus

The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus. At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth. But the definition was soon extended to [|functions of several variables] and to functions of a complex variable. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.
Functions are now used throughout all areas of mathematics. In introductory calculus, when the word function is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as real analysis and complex analysis.

Real function

A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval. In this section, these functions are simply called functions.
The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous, differentiable, and even analytic. This regularity insures that these functions can be visualized by their [|graphs]. In this section, all functions are differentiable in some interval.
Functions enjoy pointwise operations, that is, if and are functions, their sum, difference and product are functions defined by
The domains of the resulting functions are the intersection of the domains of and. The quotient of two functions is defined similarly by
but the domain of the resulting function is obtained by removing the zeros of from the intersection of the domains of and.
The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function whose graph is a hyperbola, and whose domain is the whole real line except for 0.
The derivative of a real differentiable function is a real function. An antiderivative of a continuous real function is a real function that is differentiable in any open interval in which the original function is continuous. For example, the function is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for, is a differentiable function called the natural logarithm.
A real function is monotonic in an interval if the sign of does not depend of the choice of and in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function is monotonic in an interval, it has an inverse function, which is a real function with domain and image. This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the exponential function.
Many other real functions are defined either by the implicit function theorem or as solutions of differential equations. For example, the sine and the cosine functions are the solutions of the linear differential equation
such that

Vector-valued function

When the elements of the codomain of a function are vectors, the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its velocity vector is a vector-valued function.
Some vector-valued functions are defined on a subset of or other spaces that share geometric or topological properties of, such as manifolds. These vector-valued functions are given the name vector fields.

Function space

In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support form a function space that is at the basis of the theory of distributions.
Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.

Multi-valued functions

Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a neighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point there are several possible starting values for the function.
For example, in defining the square root as the inverse function of the square function, for any positive real number there are two choices for the value of the square root, one of which is positive and denoted and another which is negative and denoted These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single smooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positive, one value for 0 and no value for negative.
In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the implicit function that maps to a root of . For one may choose either for By the implicit function theorem, each choice defines a function; for the first one, the domain is the interval and the image is ; for the second one, the domain is and the image is ; for the last one, the domain is and the image is. As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single multi-valued function of that has three values for, and only one value for and.
Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions. The domain to which a complex function may be extended by analytic continuation generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets for the square root of –1; while, when extending through complex numbers with negative imaginary parts, one gets. There are generally two ways of solving the problem. One may define a function that is not continuous along some curve, called a branch cut. Such a function is called the principal value of the function. The other way is to consider that one has a multi-valued function, which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the monodromy.

In the foundations of mathematics and set theory

The definition of a function that is given in this article requires the concept of set, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.
For example, the singleton set may be considered as a function Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definition for these weakly specified functions.
These generalized functions may be critical in the development of a formalization of the foundations of mathematics. For example, Von Neumann–Bernays–Gödel set theory, is an extension of the set theory in which the collection of all sets is a class. This theory includes the replacement axiom, which may be stated as: If is a set and is a function, then is a set.

In computer science

In computer programming, a function is, in general, a piece of a computer program, which implements the abstract concept of function. That is, it is a program unit that produces an output for each input. However, in many programming languages every subroutine is called a function, even when there is no output, and when the functionality consists simply of modifying some data in the computer memory.
Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions. For example, if_then_else is a function that takes three functions as arguments, and, depending on the result of the first function, returns the result of either the second or the third function. An important advantage of functional programming is that it makes easier program proofs, as being based on a well founded theory, the lambda calculus.
Except for computer-language terminology, "function" has the usual mathematical meaning in computer science. In this area, a property of major interest is the computability of a function. For giving a precise meaning to this concept, and to the related concept of algorithm, several models of computation have been introduced, the old ones being general recursive functions, lambda calculus and Turing machine. The fundamental theorem of computability theory is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The Church–Turing thesis is the claim that every philosophically acceptable definition of a computable function defines also the same functions.
General recursive functions are partial functions from integers to integers that can be defined from
via the operators
Although defined only for functions from integers to integers, they can model any computable function as a consequence of the following properties:
Lambda calculus is a theory that defines computable functions without using set theory, and is the theoretical background of functional programming. It consists of terms that are either variables, function definitions, or applications of functions to terms. Terms are manipulated through some rules,, which are the axioms of the theory and may be interpreted as rules of computation.
In its original form, lambda calculus does not include the concepts of domain and codomain of a function. Roughly speaking, they have been introduced in the theory under the name of type in typed lambda calculus. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus.