Partial function: Difference between revisions

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A partial function is often used when its exact ___domain of definition is not known or difficult to specify. 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. For this reason, in calculus, and more generally in [[mathematical analysis]], a partial function is generally called simply 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.

WhenFor [[Functiona (mathematics)#Arrowgiven notation|arrow<math>S notation]]\subseteq is used for functionsX</math>, a partial function <math>f</math> from <math>XS</math> to <math>Y</math> is sometimes written as <math>f : XS \rightharpoonup Y,</math> <math>f : XS \nrightarrow Y,</math> or <math>f : X S\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 : XS \rightharpoonup Y,</math> and any <math>x \in XS,</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.