Editing Function (mathematics) (section)

== Definition ==

[[File:Function machine2.svg|thumb|right|Schematic depiction of a function described metaphorically as a "machine" or "[[black box]]" that for each input yields a corresponding output]]
[[Image:Example Function.png|thumb|right|The red curve is the [[graph of a function]], because any [[Vertical line test|vertical line]] has exactly one crossing point with the curve.]]

A '''function''' {{mvar|f}} from a [[set (mathematics)|set]] {{mvar|X}} to a set {{mvar|Y}} is an assignment of one element of {{mvar|Y}} to each element of {{mvar|X}}. The set {{mvar|X}} is called the [[Domain of a function|domain]] of the function and the set {{mvar|Y}} is called the [[codomain]] of the function.

If the element {{mvar|y}} in {{mvar|Y}} is assigned to {{mvar|x}} in {{mvar|X}} by the function {{mvar|f}}, one says that {{mvar|f}} ''maps'' {{mvar|x}} to {{mvar|y}}, and this is commonly written <math>y=f(x).</math> In this notation, {{mvar|x}} is the ''[[Argument of a function|argument]]'' or ''[[Variable (mathematics)|variable]]'' of the function.

A specific element {{mvar|x}} of {{mvar|X}} is a ''value of the variable'', and the corresponding element of {{mvar|Y}} is the ''value of the function'' at {{mvar|x}}, or the [[Image (mathematics)|image]] of {{mvar|x}} under the function. The ''image of a function'', sometimes called its [[range of a function|range]], is the set of the images of all elements in the domain.<ref name="EOM Function"/><ref name="T&K Calc p.3">{{Taalman Kohn Calculus|p=3}}</ref><ref name="Trench RA pp.30-32">{{Trench Intro Real Analysis|pp=30–32}}</ref><ref name="TBB RA pp.A4-A5">{{Thomson Bruckner Bruckner Elementary Real Analysis|pp=A-4–A-5}}</ref>

A function {{mvar|f}}, its domain {{mvar|X}}, and its codomain {{mvar|Y}} are often specified  by the notation <math>f: X\to Y.</math> One may write <math>x\mapsto y</math> instead of <math>y=f(x)</math>, where the symbol <math>\mapsto</math> (read '[[maps to]]') is used to specify where a particular element {{mvar|x}} in the domain is mapped to by {{mvar|f}}.  This allows the definition of a function without naming. For example, the [[square function]] is the function <math>x\mapsto x^2.</math>

The domain and codomain are not always explicitly given when a function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if <math>f:\R\to\R</math> is a [[real function]], the determination of the domain of the function <math>x\mapsto 1/f(x)</math> requires knowing the [[zero of a function|zeros]] of {{mvar|f.}} This is one of the reasons for which, in [[mathematical analysis]], "a function {{nowrap|from {{mvar|X}} to {{mvar|Y}} "}} may refer to a function having a proper subset of {{mvar|X}} as a domain.<ref group="note">The true domain of such a function is often called the ''domain of definition'' of the function.</ref> For example, a "function from the reals to the reals" may refer to a [[real-valued function|real-valued]] function of a [[function of a real variable|real variable]] whose domain is a proper subset of the [[real number]]s, typically a subset that contains a non-empty [[open interval]]. Such a function is then called a [[partial function]].

A function {{mvar|f}} on a set {{mvar|S}} means a function from the domain {{mvar|S}}, without specifying a codomain. However, some authors use it as shorthand for saying that the function is {{math|''f'' : ''S'' → ''S''}}.

=== Formal definition ===

[[file:Injection keine Injektion 2a.svg|thumb|Diagram of a function]]
[[file:Injection keine Injektion 1.svg|thumb|Diagram of a relation that is not a function. One reason is that 2 is the first element in more than one ordered pair.  Another reason is that neither 3 nor 4 are the first element (input) of any ordered pair.]]

The above definition of a function is essentially that of the founders of [[calculus]], [[Leibniz]], [[Isaac Newton|Newton]] and [[Euler]]. However, it cannot be [[formal proof|formalized]], since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of [[set theory]]. This set-theoretic definition is based on the fact that a function establishes a ''relation'' between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, a [[binary relation]] between two sets {{math|''X''}} and {{math|''Y''}} is a [[subset]] of the set of all [[ordered pair]]s <math>(x, y)</math> such that <math>x\in X</math> and <math>y\in Y.</math> The set of all these pairs is called the [[Cartesian product]] of {{math|''X''}} and {{math|''Y''}} and denoted <math>X\times Y.</math> Thus, the above definition may be formalized as follows.

A ''function'' with domain {{math|''X''}} and codomain {{math|''Y''}} is a binary relation {{mvar|R}} between {{math|''X''}} and {{math|''Y''}} that satisfies the two following conditions:<ref>{{cite book | last=Halmos | first=Paul R. | title=Naive Set Theory | publisher=Springer | year=1974 | pages=30–33}}</ref>
* For every <math>x</math> in <math>X</math> there exists <math>y</math> in <math>Y</math> such that <math>(x,y)\in R.</math>
* If <math>(x,y)\in R</math> and <math>(x,z)\in R,</math> then <math>y=z.</math>

This definition may be rewritten more formally, without referring explicitly to the concept of a relation, but using more notation (including [[set-builder notation]]):

A function is formed by three sets, the ''domain'' <math>X,</math> the ''codomain'' <math>Y,</math> and the ''graph'' <math>R</math> that satisfy the three following conditions. 
*<math>R \subseteq \{(x,y) \mid x\in X, y\in Y\}</math>
*<math>\forall x\in X, \exists y\in Y, \left(x, y\right) \in R \qquad</math>
*<math>(x,y)\in R \land (x,z)\in R \implies y=z\qquad</math>

=== Partial functions ===
{{main|Partial function}}

Partial functions are defined similarly to ordinary functions, with the "total" condition removed. That is, a ''partial function'' from {{mvar|X}} to {{mvar|Y}} is a binary relation {{mvar|R}} between {{mvar|X}} and {{mvar|Y}} such that, for every <math>x\in X,</math> there is ''at most one'' {{mvar|y}} in {{mvar|Y}} such that <math>(x,y) \in R.</math>

Using functional notation, this means that, given <math>x\in X,</math> either <math>f(x)</math> is in {{mvar|Y}}, or it is undefined.

The set of the elements of {{mvar|X}} such that <math>f(x)</math> is defined and belongs to {{mvar|Y}} is called the ''domain of definition'' of the function. A partial function from {{mvar|X}} to {{mvar|Y}} is thus an ordinary function that has as its domain a subset of {{mvar|X}} called the domain of definition of the function. If the domain of definition equals {{mvar|X}}, one often says that the partial function is a ''total function''.

In several areas of mathematics, the term "function" refers to partial functions rather than to ordinary (total) functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain.

In [[calculus]], a ''real-valued function of a real variable'' or ''[[real function]]'' is a partial function from the set <math>\R</math> of the [[real number]]s to itself. Given a real function <math>f:x\mapsto f(x)</math> its [[multiplicative inverse]] <math>x\mapsto 1/f(x)</math> is also a real function. The determination of the domain of definition of a multiplicative inverse of a (partial) function amounts to compute the [[zero of a function|zeros]] of the function, the values where the function is defined but not its multiplicative inverse.

Similarly, a ''[[function of a complex variable]]'' is generally a partial function whose domain of definition is a subset of the [[complex number]]s <math>\Complex</math>. The difficulty of determining the domain of definition of a [[complex function]] is illustrated by the multiplicative inverse of the [[Riemann zeta function]]: the determination of the domain of definition of the function <math>z\mapsto 1/\zeta(z)</math> is more or less equivalent to the proof or disproof of one of the major open problems in mathematics, the [[Riemann hypothesis]]. 

In [[computability theory]], a [[general recursive function]] is a partial function from the integers to the integers whose values can be computed by an [[algorithm]] (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether {{math|0}} belongs to its domain of definition (see [[Halting problem]]).

=== Multivariate functions <span class="anchor" id="MULTIVARIATE_FUNCTION"></span> ===
{{distinguish|Multivalued function}}
[[File:Binary operations as black box.svg|thumb|A binary operation is a typical example of a bivariate function which assigns to each pair <math>(x, y)</math> the result <math>x\circ y</math>.]]

A '''multivariate function''', '''multivariable 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 travelled and its average speed.

Formally, a function of {{mvar|n}} variables is a function whose domain is a set of {{mvar|n}}-tuples.<ref group=note>{{mvar|n}} may also be 1, thus subsuming functions as defined above. For {{math|1=''n'' = 0}}, each [[constant (mathematics)|constant]] is a special case of a multivariate function, too.</ref> For example, multiplication of [[integer]]s is a function of two variables, or '''bivariate function''', whose domain is the set of all [[ordered pairs]] (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every [[binary operation]]. The graph of a bivariate surface over a two-dimensional real domain may be interpreted as defining a [[Surface (mathematics)#Graph of a bivariate function|parametric surface]], as used in, e.g., [[bivariate interpolation]].

Commonly, an {{mvar|n}}-tuple is denoted enclosed between parentheses, such as in <math>(1,2,\ldots, n).</math>  When using [[functional notation]], one usually omits the parentheses surrounding tuples, writing  <math>f(x_1,\ldots,x_n)</math> instead of <math>f((x_1,\ldots,x_n)).</math>

Given {{mvar|n}} sets <math>X_1,\ldots, X_n,</math> the set of all {{mvar|n}}-tuples <math>(x_1,\ldots,x_n)</math> such that <math>x_1\in X_1, \ldots, x_n\in X_n</math> is called the [[Cartesian product]] of <math>X_1,\ldots, X_n,</math> and denoted <math>X_1\times\cdots\times X_n.</math>

Therefore, a multivariate function is a function that has a Cartesian product or a [[proper subset]] of a Cartesian product as a domain.

<math display="block">f: U\to Y,</math>

where the domain {{mvar|U}} has the form

<math display="block">U\subseteq X_1\times\cdots\times X_n.</math>

If all the <math>X_i</math> are equal to the set <math>\R</math> of the [[real number]]s or to the set <math>\C</math> of the [[complex number]]s, one talks respectively of a [[function of several real variables]] or of a [[function of several complex variables]].