# Identity function

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Function that returns its argument unchanged

Not to be confused with [Null function](/source/Null_function) or [Empty function](/source/Empty_function).

[Graph](/source/Graph_of_a_function) of the identity function on the [real numbers](/source/Real_number)

In [mathematics](/source/Mathematics), an **identity function**, also called an **identity relation**, **identity map** or **identity transformation**, is a [function](/source/Function_(mathematics)) that always returns the value that was used as its [argument](/source/Argument_of_a_function), unchanged. That is, when f {\displaystyle f} is the identity function, the [equality](/source/Equality_(mathematics)) f ( x ) = x {\displaystyle f(x)=x} is true for all values of x {\displaystyle x} to which f {\displaystyle f} can be applied.

## Definition

Formally, if X {\displaystyle X} is a [set](/source/Set_(mathematics)), the identity function f {\displaystyle f} on X {\displaystyle X} is defined to be a function with X {\displaystyle X} as its [domain](/source/Domain_of_a_function) and [codomain](/source/Codomain), satisfying

        f
        (
        x
        )
        =
        x

    {\displaystyle f(x)=x}

 for all elements

        x

    {\displaystyle x}

 in

        X

    {\displaystyle X}

.[1]

In other words, the function value f ( x ) {\displaystyle f(x)} in the codomain X {\displaystyle X} is always the same as the input element x {\displaystyle x} in the domain X {\displaystyle X} . The identity function on X {\displaystyle X} is clearly an [injective function](/source/Injective_function) as well as a [surjective function](/source/Surjective_function) (its codomain is also its [range](/source/Range_(function))), so it is [bijective](/source/Bijection).[2]

The identity function f {\displaystyle f} on X {\displaystyle X} is often denoted by i d X {\displaystyle \mathrm {id} _{X}} .

In [set theory](/source/Set_theory), where a function is defined as a particular kind of [binary relation](/source/Binary_relation), the identity function is given by the [identity relation](/source/Identity_relation), or *diagonal* of X {\displaystyle X} .[3]

## Algebraic properties

If f : X → Y {\displaystyle f:X\rightarrow Y} is any function, then f ∘ i d X = f = i d Y ∘ f {\displaystyle f\circ \mathrm {id} _{X}=f=\mathrm {id} _{Y}\circ f} , where " ∘ {\displaystyle \circ } " denotes [function composition](/source/Function_composition).[4] In particular, i d X {\displaystyle \mathrm {id} _{X}} is the [identity element](/source/Identity_element) of the [monoid](/source/Monoid) of all functions from X {\displaystyle X} to X {\displaystyle X} (under function composition).

Since the identity element of a monoid is [unique](/source/Unique_(mathematics)),[5] one can alternately define the identity function on M {\displaystyle M} to be this identity element. Such a definition generalizes to the concept of an [identity morphism](/source/Identity_morphism) in [category theory](/source/Category_theory), where the [endomorphisms](/source/Endomorphism) of M {\displaystyle M} need not be functions.

## Properties

- The identity function is a [linear operator](/source/Linear_map) when applied to [vector spaces](/source/Vector_space).[6]

- In an n {\displaystyle n} -[dimensional](/source/Dimension_(vector_space)) [vector space](/source/Vector_space) the identity function is represented by the [identity matrix](/source/Identity_matrix) I n {\displaystyle I_{n}} , regardless of the [basis](/source/Basis_(linear_algebra)) chosen for the space.[7]

- The identity function on the positive [integers](/source/Integer) is a [completely multiplicative function](/source/Completely_multiplicative_function) (essentially multiplication by 1), considered in [number theory](/source/Number_theory).[8]

- In a [metric space](/source/Metric_space) the identity function is trivially an [isometry](/source/Isometry). An object without any [symmetry](/source/Symmetry) has as its [symmetry group](/source/Symmetry_group) the [trivial group](/source/Trivial_group) containing only this isometry (symmetry type C 1 {\displaystyle \mathrm {C} _{1}} ).[9]

- In a [topological space](/source/Topological_space), the identity function is always [continuous](/source/Continuous_function#Continuous_functions_between_topological_spaces).[10]

- The identity function is [idempotent](/source/Idempotence).[11]

- Every map from a [set of a single element](/source/Singleton_(mathematics)) to itself is necessarily the identity map.

## See also

- [Identity matrix](/source/Identity_matrix)

- [Inclusion map](/source/Inclusion_map)

- [Indicator function](/source/Indicator_function)

## References

1. **[^](#cite_ref-1)** Knapp, Anthony W. (2006). *Basic algebra*. Springer. [ISBN](/source/ISBN_(identifier)) [978-0-8176-3248-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8176-3248-9).

1. **[^](#cite_ref-2)** Mapa, Sadhan Kumar (7 April 2014). *Higher Algebra Abstract and Linear* (11th ed.). Sarat Book House. p. 36. [ISBN](/source/ISBN_(identifier)) [978-93-80663-24-1](https://en.wikipedia.org/wiki/Special:BookSources/978-93-80663-24-1).

1. **[^](#cite_ref-3)** [*Proceedings of Symposia in Pure Mathematics*](https://books.google.com/books?id=oIFLAQAAIAAJ&q=the+identity+function+is+given+by+the+identity+relation,+or+diagonal). American Mathematical Society. 1974. p. 92. [ISBN](/source/ISBN_(identifier)) [978-0-8218-1425-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-1425-3). ...then the diagonal set determined by M is the identity relation...

1. **[^](#cite_ref-4)** Nel, Louis (2016). [*Continuity Theory*](https://books.google.com/books?id=_JdPDAAAQBAJ&pg=PA21). Cham: Springer. p. 21. [doi](/source/Doi_(identifier)):[10.1007/978-3-319-31159-3](https://doi.org/10.1007%2F978-3-319-31159-3). [ISBN](/source/ISBN_(identifier)) [978-3-319-31159-3](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-31159-3).

1. **[^](#cite_ref-5)** Rosales, J. C.; García-Sánchez, P. A. (1999). [*Finitely Generated Commutative Monoids*](https://books.google.com/books?id=LQsH6m-x8ysC&q=identity+element+of+a+monoid+is+unique&pg=PA1). Nova Publishers. p. 1. [ISBN](/source/ISBN_(identifier)) [978-1-56072-670-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-56072-670-8). The element 0 is usually referred to as the identity element and if it exists, it is unique

1. **[^](#cite_ref-6)** Anton, Howard (2005), *Elementary Linear Algebra (Applications Version)* (9th ed.), Wiley International

1. **[^](#cite_ref-7)** T. S. Shores (2007). [*Applied Linear Algebra and Matrix Analysis*](https://books.google.com/books?id=8qwTb9P-iW8C&q=Matrix+Analysis). Undergraduate Texts in Mathematics. Springer. [ISBN](/source/ISBN_(identifier)) [978-038-733-195-9](https://en.wikipedia.org/wiki/Special:BookSources/978-038-733-195-9).

1. **[^](#cite_ref-8)** D. Marshall; E. Odell; M. Starbird (2007). *Number Theory through Inquiry*. Mathematical Association of America Textbooks. Mathematical Assn of Amer. [ISBN](/source/ISBN_(identifier)) [978-0883857519](https://en.wikipedia.org/wiki/Special:BookSources/978-0883857519).

1. **[^](#cite_ref-9)** Anderson, James W. (2007). *Hyperbolic geometry*. Springer undergraduate mathematics series (2. ed., corr. print ed.). London: Springer. [ISBN](/source/ISBN_(identifier)) [978-1-85233-934-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-85233-934-0).

1. **[^](#cite_ref-10)** Conover, Robert A. (2014-05-21). [*A First Course in Topology: An Introduction to Mathematical Thinking*](https://books.google.com/books?id=KCziAgAAQBAJ&q=identity+function+is+always+continuous&pg=PA65). Courier Corporation. p. 65. [ISBN](/source/ISBN_(identifier)) [978-0-486-78001-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-78001-6).

1. **[^](#cite_ref-11)** Conferences, University of Michigan Engineering Summer (1968). [*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.

v t e Function History List of specific functions Types by domain, codomain X → 𝔹 𝔹 → X 𝔹ⁿ → 𝔹 X → ℤ ℤ → X X → ℝ ℝ → X ℝⁿ → X X → ℂ ℂ → X ℂⁿ → X Classes, properties Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective Constructions Restriction Composition λ Inverse Generalizations Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor Category

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Adapted from the Wikipedia article [Identity function](https://en.wikipedia.org/wiki/Identity_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Identity_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.