Identity function

Algebraic properties

If ${\displaystyle f:X\to Y}$ is any function, then ${\displaystyle f\circ {\mathrm{i}\mathrm{d}}_X=f={\mathrm{i}\mathrm{d}}_Y\circ f}$, where "${\displaystyle \circ }$" denotes function composition. In particular, ${\displaystyle {\mathrm{i}\mathrm{d}}_X}$ is the identity element of the monoid of all functions from ${\displaystyle X}$ to ${\displaystyle X}$ (under function composition).

Since the identity element of a monoid is unique, one can alternately define the identity function on ${\displaystyle M}$ to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of ${\displaystyle M}$ need not be functions.

Properties

• The identity function is a linear operator when applied to vector spaces.
• In an ${\displaystyle n}$-dimensional vector space the identity function is represented by the identity matrix ${\displaystyle I_n}$, 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 ${\displaystyle {\mathrm{C}}_1}$).
• In a topological space, the identity function is always continuous.
• The identity function is idempotent.
• Every map from a set of a single element to itself is necessarily the identity map.