# Identity function

__: In mathematics, a function that always returns the same value that was used as its argument__

**Short description**thumb|[[Graph of a function|Graph of the identity function on the real numbers]]

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.

## Definition

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

In other words, the function value *f*(*X*) in the codomain *M* is always the same as the input element *X* in the domain *M*. The identity function on M is clearly an injective function as well as a surjective function, so it is 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* (under function composition).

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]} - 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.^{[6]} - The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
^{[7]} - 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}).^{[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. - ↑ (in en)
*Proceedings of Symposia in Pure Mathematics*. American Mathematical Society. 1974. pp. 92. ISBN 978-0-8218-1425-3. https://books.google.com/books?id=oIFLAQAAIAAJ&q=the+identity+function+is+given+by+the+identity+relation,+or+diagonal. "...then the diagonal set determined by M is the identity relation..." - ↑ Rosales, J. C.; García-Sánchez, P. A. (1999) (in en).
*Finitely Generated Commutative Monoids*. Nova Publishers. pp. 1. ISBN 978-1-56072-670-8. https://books.google.com/books?id=LQsH6m-x8ysC&q=identity+element+of+a+monoid+is+unique&pg=PA1. "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. https://books.google.com/books?id=8qwTb9P-iW8C&q=Matrix+Analysis. - ↑ 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) (in en).
*A First Course in Topology: An Introduction to Mathematical Thinking*. Courier Corporation. pp. 65. ISBN 978-0-486-78001-6. https://books.google.com/books?id=KCziAgAAQBAJ&q=identity+function+is+always+continuous&pg=PA65. - ↑ Conferences, University of Michigan Engineering Summer (1968) (in en).
*Foundations of Information Systems Engineering*. https://books.google.com/books?id=AvAfAAAAMAAJ&q=The+identity+function+is+idempotent.. "we see that an identity element of a semigroup is idempotent."

Original source: https://en.wikipedia.org/wiki/Identity function.
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