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Identity function

Function that returns its argument unchanged

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Source updated March 2, 2026

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.

01Definition

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 $\mathrm {id} _{X}$ .

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 $X$ .

02Algebraic properties

If $f:X\rightarrow Y$ is any function, then $f\circ \mathrm {id} _{X}=f=\mathrm {id} _{Y}\circ f$ , where " $\circ$ " denotes function composition. In particular, $\mathrm {id} _{X}$ 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.

03Properties

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 $\mathrm {C} _{1}$ ).
In a topological space, the identity function is always continuous.
The identity function is idempotent.
Every map from a set of a single element to itself is necessarily the identity map.

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Sources and credits

This article is adapted from the Wikipedia article “Identity function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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