Editing Partial function (section)

{{distinguish|text=the [[partial application]] of a function of several variables, by fixing some of them}}
{{short description|Function whose actual domain of definition may be smaller than its apparent domain}}
{{more footnotes|date=August 2014}}

In [[mathematics]], a '''partial function''' {{mvar|f}} from a [[Set (mathematics)|set]] {{mvar|X}} to a set {{mvar|Y}} is a [[function (mathematics)|function]] from a [[subset]] {{mvar|S}} of {{mvar|X}} (possibly the whole {{mvar|X}} itself) to {{mvar|Y}}. The subset {{mvar|S}}, that is, the ''[[Domain of a function|domain]]'' of {{mvar|f}} viewed as a function, is called the '''domain of definition''' or '''natural domain''' of {{mvar|f}}. If {{mvar|S}} equals {{mvar|X}}, that is, if {{mvar|f}} is defined on every element in {{mvar|X}}, then {{mvar|f}} is said to be a '''total function'''.

In other words, a partial function is a [[binary relation]] over two [[Set (mathematics)|sets]] that associates to every element of the first set ''at most'' one element of the second set; it is thus a [[univalent relation]]. This generalizes the concept of a (total) [[Function (mathematics)|function]] by not requiring ''every'' element of the first set to be associated to an element of the second set.

A partial function is often used when its exact domain of definition is not known, or is difficult to specify. However, even when the exact domain of definition is known, partial functions are often used for simplicity or brevity. This is the case in [[calculus]], where, for example, the [[quotient]] of two functions is a partial function whose domain of definition cannot contain the [[Zero of a function|zeros]] of the denominator; in this context, a partial function is generally simply called a {{em|function}}.

In [[computability theory]], a [[general recursive function]] is a partial function from the integers to the integers; no [[algorithm]] can exist for deciding whether an arbitrary such function is in fact total.

When [[Function (mathematics)#Arrow notation|arrow notation]] is used for functions, a partial function <math>f</math> from <math>X</math> to <math>Y</math> is sometimes written as <math>f : X \rightharpoonup Y,</math> <math>f : X \nrightarrow Y,</math> or <math>f : X \hookrightarrow Y.</math> However, there is no general convention, and the latter notation is more commonly used for [[inclusion map]]s or [[embedding]]s.{{citation needed|reason=Provide a few example citations for each notation.|date=July 2019}}

Specifically, for a partial function <math>f : X \rightharpoonup Y,</math> and any <math>x \in X,</math> one has either:
* <math>f(x) = y \in Y</math> (it is a single element in {{mvar|Y}}), or
* <math>f(x)</math> is undefined.

For example, if <math>f</math> is the [[square root]] function restricted to the [[integer]]s
: <math>f : \Z \to \N,</math> defined by:
: <math>f(n) = m</math> if, and only if, <math>m^2 = n,</math> <math>m \in \N, n \in \Z,</math>
then <math>f(n)</math> is only defined if <math>n</math> is a [[Square number|perfect square]] (that is, <math>0, 1, 4, 9, 16, \ldots</math>). So <math>f(25) = 5</math> but <math>f(26)</math> is undefined.