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.

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

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]}

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.

- 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 C
_{1}).^{[9]} - In a topological space, the identity function is always continuous.
^{[10]} - The identity function is idempotent.
^{[11]}

**^**Knapp, Anthony W. (2006),*Basic algebra*, Springer, ISBN 978-0-8176-3248-9**^**Mapa, Sadhan Kumar (7 April 2014).*Higher Algebra Abstract and Linear*(11th ed.). Sarat Book House. p. 36. ISBN 978-93-80663-24-1.**^***Proceedings of Symposia in Pure Mathematics*. American Mathematical Society. 1974. p. 92. ISBN 978-0-8218-1425-3....then the diagonal set determined by M is the identity relation...

**^**Nel, Louis (2016).*Continuity Theory*. p. 21. doi:10.1007/978-3-319-31159-3. ISBN 978-3-319-31159-3.**^**Rosales, J. C.; García-Sánchez, P. A. (1999).*Finitely Generated Commutative Monoids*. Nova Publishers. p. 1. ISBN 978-1-56072-670-8.The element 0 is usually referred to as the identity element and if it exists, it is unique

**^**Anton, Howard (2005),*Elementary Linear Algebra (Applications Version)*(9th ed.), Wiley International**^**T. S. Shores (2007).*Applied Linear Algebra and Matrix Analysis*. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.**^**D. Marshall; E. Odell; M. Starbird (2007).*Number Theory through Inquiry*. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.**^**James W. Anderson,*Hyperbolic Geometry*, Springer 2005, ISBN 1-85233-934-9**^**Conover, Robert A. (2014-05-21).*A First Course in Topology: An Introduction to Mathematical Thinking*. Courier Corporation. p. 65. ISBN 978-0-486-78001-6.**^**Conferences, University of Michigan Engineering Summer (1968).*Foundations of Information Systems Engineering*.we see that an identity element of a semigroup is idempotent.