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Not to be confused with Null function. This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009) In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument. In terms of equations, the function is given by f(x) = x. Contents 1 Definition 2 Algebraic property 3 Examples 4 See also // Definition Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies f(x) = x    for all elements x in M. In other words, the function assigns to each element x of M the element x of M. The identity function f on M is often denoted by idM or 1M. In terms of 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 M. Algebraic property If f : M → N is any function, then we have f o idM = f = idN o f (where "o" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M. 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. Examples 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 n-dimensional vector space the identity function is represented by the identity matrix In, 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 C1). See also Inclusion map