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Partial function

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

Not to be confused with the partial application of a function of several variables, by fixing some of them.

More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a univalent relation. This generalizes the concept of a (total) 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 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 zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a 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 arrow notation is used for functions, a partial function from to is sometimes written as or However, there is no general convention, and the latter notation is more commonly used for inclusion maps or embeddings.


Specifically, for a partial function and any one has either:


For example, if is the square root function restricted to the integers


then is only defined if is a perfect square (that is, ). So but is undefined.

 – Extension of the domain of an analytic function (mathematics)

Analytic continuation

 – Generalized mathematical function

Multivalued function

 – Function that is defined almost everywhere (mathematics)

Densely defined operator

(1958), Computability and Unsolvability, McGraw–Hill Book Company, Inc, New York. Republished by Dover in 1982. ISBN 0-486-61471-9.

Martin Davis

(1952), Introduction to Meta-Mathematics, North-Holland Publishing Company, Amsterdam, Netherlands, 10th printing with corrections added on 7th printing (1974). ISBN 0-7204-2103-9.

Stephen Kleene

(1972), Introduction to Computer Organization and Data Structures, McGraw–Hill Book Company, New York.

Harold S. Stone