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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.

More technically, a partial function is a binary relation over two 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 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 zeros of the denominator; in this context, a partial function is generally simply called 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 $f$ from $X$ to $Y$ is sometimes written as $f:X\rightharpoonup Y,$ $f:X\nrightarrow Y,$ or $f:X\hookrightarrow Y.$ However, there is no general convention, and the latter notation is more commonly used for inclusion maps or embeddings.

Specifically, for a partial function $f:X\rightharpoonup Y,$ and any $x\in X,$ one has either:

• $f\left(x\right)=y\in Y$ (it is a single element in Y), or
• $f\left(x\right)$ is undefined.

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

$f:Z\to N,$ defined by:

$f\left(n\right)=m$ if, and only if, $m^2=n,$ $m\in N,n\in Z,$

then $f\left(n\right)$ is only defined if $n$ is a perfect square (that is, $0,1,4,9,16,\dots$). So $f\left(25\right)=5$ but $f\left(26\right)$ is undefined.