Graph of the identity function on the real numbers
, 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. That is, for being identity, the equality
holds for all .
Formally, if is a set
, the identity function on is defined to be that function with domain
: for all elements in .
In other words, the function value in (that is, the codomain) is always the same input element of (now considered as the domain). The identity function on is clearly an injective function
as well as a surjective function
, so it is also 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 .
If is any function, then we have (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 endomorphism
s of need not be functions.
*The identity function is a linear operator
, when applied to vector space
*The identity function on the positive integer
s is a completely multiplicative function
(essentially multiplication by 1), considered in number theory
*In an -dimensional vector space
the identity function is represented by the identity matrix
, regardless of the basis
*In a metric space
the identity is trivially an isometry
. An object without any symmetry
has as symmetry group
the trivial group only containing this isometry (symmetry type ).
*In a topological space
, the identity function is always continuous.
*The identity function is idempotent
Category:Functions and mappings
Category:Basic concepts in set theory
Category:Types of functions