In mathematics, an **identity function**, also called an **identity relation** or **identity map** or **identity transformation**, is a function that always returns the same value that was used as its argument. That is, for f being identity, the equality *f*(*x*) = *x* holds for all x.

## Definition

Formally, if *M* is a set, the identity function *f* on *M* is defined to be that function with domain and codomain *M* which satisfies

*f*(*x*) =*x*for all elements*x*in*M*.^{[1]}

In other words, the function value *f*(*x*) in *M* (that is, the codomain) is always the same input element *x* of *M* (now considered as the domain). The identity function on M is clearly an injective function as well as a surjective function, so it is also bijective.^{[2]}

The identity function *f* on *M* is often denoted by id_{M}.

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 *M*.^{[3]}

## Algebraic properties

If *f* : *M* → *N* is any function, then we have *f* ∘ id_{M} = *f* = id_{N} ∘ *f* (where "∘" denotes function composition). In particular, id_{M} is the identity element of the monoid of all functions from *M* to *M*.

Since the identity element of a monoid is unique,^{[4]} 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.
^{[5]} - The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
^{[6]} - In an
*n*-dimensional vector space the identity function is represented by the identity matrix I_{n}, regardless of the basis.^{[7]} - In a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the trivial group only containing this isometry (symmetry type
*C*_{1}).^{[8]} - In a topological space, the identity function is always continuous.
^{[9]} - The identity function is idempotent.
^{[10]}

## See also

## References

**^**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...

**^**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**^**D. Marshall; E. Odell; M. Starbird (2007).*Number Theory through Inquiry*. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.**^**T. S. Shores (2007).*Applied Linear Algebra and Matrix Analysis*. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.**^**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.