Математикадағы өзіндік функция

In mathematics, a function that always returns the same value that was used as its argument

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 1=f(x) = x is true for all values of x to which f can be applied.

If f : X → Y is any function, then 1=f ∘ idX = f = idY ∘ f, where "∘" denotes function composition. In particular, idX is the identity element of the monoid of all functions from X to X (under function composition). 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.

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 In, 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 C1). In a topological space, the identity function is always continuous. The identity function is idempotent.