Identity function

Not to be confused with Null function or Empty 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 f is the identity function, the equality f(x) = x is true for all values of x to which f can be applied.

Graph of the identity function on the real numbers

Definition

Formally, if X is a set, the identity function f on X is defined to be a function with X as its domain and codomain, satisfying

f(x) = x   for all elements x in X.^[1]

In other words, the function value f(x) in the codomain X is always the same as the input element x in the domain X. The identity function on X is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.^[2]

The identity function f on X is often denoted by id_X.

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 X.^[3]

Algebraic properties

If f : X → Y is any function, then f ∘ id_X = f = id_Y ∘ f, where "∘" denotes function composition.^[4] In particular, id_X 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,^[5] 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.

Properties

• The identity function is a linear operator when applied to vector spaces.^[6]
• In an n-dimensional vector space the identity function is represented by the identity matrix I_n, regardless of the basis chosen for the space.^[7]
• The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.^[8]
• 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).^[9]
• In a topological space, the identity function is always continuous.^[10]
• The identity function is idempotent.^[11]

See also

• Identity matrix
• Inclusion map
• Indicator function