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- Identity function

thumb|Graph of the identity function on the real numbers

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 being identity, the equality holds for all .

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

for all elements in .

In other words, the function value in (that is, the codomain) is always the same input element of (now considered as the domain). The identity function on is clearly an injective function as well as a surjective function, so it is also bijective.^{[1]}

The identity function on is often denoted by .

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 .^{[2]}

If is any function, then we have (where "∘" denotes function composition). In particular, is the identity element of the monoid of all functions from to .

Since the identity element of a monoid is unique,^{[3]} one can alternately define the identity function on to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of need not be functions.

- The identity function is a linear operator, when applied to vector spaces.
- The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
^{[4]} - In an -dimensional vector space the identity function is represented by the identity matrix, regardless of the basis.
^{[5]} - 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).
^{[6]} - In a topological space, the identity function is always continuous.
^{[7]} - The identity function is idempotent.
^{[8]}

- Book: Mapa, Sadhan Kumar . 7 April 2014. Higher Algebra Abstract and Linear . 11th . Sarat Book House . 36 . 978-93-80663-24-1.
- Book: Proceedings of Symposia in Pure Mathematics. 1974. American Mathematical Society. 978-0-8218-1425-3. 92. en. ...then the diagonal set determined by M is the identity relation....
- Book: Rosales. J. C.. Finitely Generated Commutative Monoids. García-Sánchez. P. A.. 1999. Nova Publishers. 978-1-56072-670-8. 1. en. The element 0 is usually referred to as the identity element and if it exists, it is unique.
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- Book: Applied Linear Algebra and Matrix Analysis. T. S. Shores. 2007. Springer. 978-038-733-195-9. Undergraduate Texts in Mathematics.
- ,
*Hyperbolic Geometry*, Springer 2005, - Book: Conover, Robert A.. A First Course in Topology: An Introduction to Mathematical Thinking. 2014-05-21. Courier Corporation. 978-0-486-78001-6. 65. en.
- Book: Conferences, University of Michigan Engineering Summer. Foundations of Information Systems Engineering. 1968. en. we see that an identity element of a semigroup is idempotent..