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

In other words, the function value $f(x) = x$ in the codomain $x$ is always the same as the input element $X$ in the domain $f(x)$. 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 $X$ on $x$ is often denoted by $X$.

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 $f$.

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

Since the identity element of a monoid is unique, one can alternately define the identity function on $id_{X}$ to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of $X$ 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 $X$, 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 $M$).
 * In a topological space, the identity function is always continuous.
 * The identity function is idempotent.