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More technically, a partial function is a [[binary relation]] over two [[Set (mathematics)|sets]] that associates every element of the first set to ''at most'' one element of the second set; it is thus a [[Binary relation#Special types of binary relations|functional binary relation]]. It generalizes the concept of a (total) [[Function (mathematics)|function]] by not requiring every element of the first set to be associated to ''exactly'' one element of the second set.
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;
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}}
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