# Identity function

(Redirected from Identity map)
Not to be confused with Null function or Empty function.
Graph of the identity function on the real numbers

In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In terms of equations, the function is given by f(x) = x.

## Definition

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

f(x) = x    for all elements x in M.[1]

In other words, the function assigns to each element x of M the element x of M.

The identity function f on M is often denoted by idM.

In terms of set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of M.

## Algebraic property

If f : M → N is any function, then we have f o idM = f = idN o f (where "o" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M.

Since the identity element of a monoid is unique, one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.