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Graph of the identity function on the real numbers In mathematics, 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 .

Definition

Formally, if is a set, the identity function on is defined to be that function with domain and codomain which satisfies :   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 .

Algebraic properties

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 endomorphisms of need not be functions.

Properties

*The identity function is a linear operator, when applied to vector spaces. *The identity function on the positive integers 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.