What is "partial function"

n. (context mathematics English) A function whose domain is a subset of some set

An example of partial function that is injective.

An example of total function that is not injective.

In mathematics, a partial function from X to Y (written as ) is a function , for some subset X ′ of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y (only some subset X ′ of X). If , then f is called a total function and is equivalent to a function. Partial functions are often used when the exact domain, X, is not known (e.g. many functions in computability theory).

Specifically, we will say that for any , either:

• (it is defined as a single element in Y) or

• f(x) is undefined.

For example, we can consider the square root function restricted to the integers

g: Z → Z

$$g(n) = \sqrt{n}.$$

Thus g(n) is only defined for n that are perfect squares . So, , but g(26) is undefined.