# Identity function

In mathematics, an **identity function**, also called an **identity relation**, **identity map** or **identity transformation**, is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality *f*(*x*) = *x* is true for all values of x to which f can be applied.

## Definition

Formally, if *X* is a set, the identity function *f* on *X* is defined to be a function with *X* as its domain and codomain, satisfying

*f*(

*x*) =

*x*for all elements

*x*in

*X*.

In other words, the function value *f*(*x*) in the codomain *X* is always the same as the input element *x* in the domain *X*. The identity function on X is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.

The identity function *f* on *X* is often denoted by id_{X}.

In 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 *X*.

## Algebraic properties

If *f* : *X* → *Y* is any function, then *f* ∘ id_{X} = *f* = id_{Y} ∘ *f*, where "∘" denotes function composition. In particular, id_{X} is the identity element of the monoid of all functions from *X* to *X* (under function composition).

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.

## Properties

- The identity function is a linear operator when applied to vector spaces.
- In an n-dimensional vector space the identity function is represented by the identity matrix
*I*_{n}, regardless of the basis chosen for the space. - The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
- In a metric space the identity function is trivially an isometry. An object without any symmetry has as its symmetry group the trivial group containing only this isometry (symmetry type C
_{1}). - In a topological space, the identity function is always continuous.
- The identity function is idempotent.