Editing Function (mathematics) (section)

=== 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]]).