Identity function

 ( Functions And Mappings ) 

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 is the identity function, the equality is true for all values of to which can be applied.

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

In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijection.

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 .

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Algebraic properties If is any function, then , where "∘" denotes function composition.

Louis Nel (2025). 9783319311593, Springer. . ISBN 9783319311593

In particular, is the identity element of the monoid of all functions from to (under function composition).

Since the identity element of a monoid is unique,

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 of need not be functions.

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Properties

• The identity function is a linear map when applied to .
• In an -dimensional vector space the identity function is represented by the identity matrix , regardless of the basis chosen for the space.
  T. S. Shores (2025). 9780387331959, Springer. . ISBN 9780387331959
• The identity function on the positive is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
  (2025). 9780883857519, Mathematical Assn of Amer. ISBN 9780883857519
• 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 ).
  James W. Anderson (2025). 9781852339340, Springer. ISBN 9781852339340
• In a topological space, the identity function is always continuous.
  Robert A. Conover (2014). 9780486780016, Courier Corporation. . ISBN 9780486780016
• The identity function is Idempotence.

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See also

• Identity matrix
• Inclusion map
• Indicator function

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