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 $f$ is the identity function, the equality $f(x)=x$ is true for all values of $x$ to which $f$ can be applied.

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Definition 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 bijection.

The identity function $f$ on $X$ is often denoted by $\backslash mathrm\{id\}\_X$.

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 $X$.

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Algebraic properties If $f:X\backslash rightarrow\; Y$ is any function, then $f\backslash circ\backslash mathrm\{id\}\_X=f=\backslash mathrm\{id\}\_Y\backslash circ\; f$, where "$\backslash circ$" denotes function composition.

Louis Nel (2026). 9783319311593, Springer. . ISBN 9783319311593

In particular, $\backslash mathrm\{id\}\_X$ 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,

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 of $M$ need not be functions.

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Properties

• The identity function is a linear map when applied to .
• 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.
  T. S. Shores (2026). 9780387331959, Springer. . ISBN 9780387331959
• The identity function on the positive is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
  (2026). 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 $\backslash mathrm\{C\}\_1$).
  James W. Anderson (2026). 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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