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 is the identity function, the equality is true for all values of to which can be applied.

Definition

Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying
In other words, the function value in the codomain is always the same as the input element in the domain. The identity function on is clearly an injective function as well as a surjective function, so it is bijective.
The identity function on is often denoted by.
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.

Algebraic properties

If is any function, then, where "" denotes function composition. In particular, is the identity element of the monoid of all functions from to .
Since the identity element of a monoid is unique, one can alternately define the identity function on to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of need not be functions.

Properties

The identity function is a linear operator when applied to vector spaces.
In an -dimensional vector space the identity function is represented by the identity matrix, regardless of the basis chosen for the space.
The identity function on the positive integers is a completely multiplicative function, 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.
In a topological space, the identity function is always continuous.
The identity function is idempotent.