Identity function

The function id_A:A→A defined by id_A(a)=a

Identity function

For a set $A$ , the identity function on $A$ is the function

\mathrm{id}_A\colon A\to A,\qquad \mathrm{id}_A(a)=a.

The identity function is bijective . Identity functions are characterized by their behavior under composition : for any $f\colon A\to B$ , one has $\mathrm{id}_B\circ f=f$ and $f\circ \mathrm{id}_A=f$ .

Examples:

On $\mathbb{R}$ , $\mathrm{id}_{\mathbb{R}}(x)=x$ .
On a finite set $A=\{1,2,3\}$ , $\mathrm{id}_A$ fixes each element.