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{{Short description|Function that returns its argument unchanged}}
{{distinguish|Null function|Empty function}}
[[image:Function-x.svg|thumb|[[Graph of a function|Graph]] of the identity function on the [[real number]]s]]

In [[mathematics]], an '''identity function''', also called an '''identity relation''', '''identity map''' or '''identity transformation''', is a [[function (mathematics)|function]] that always returns the value that was used as its [[argument of a function|argument]], unchanged. That is, when {{mvar|f}} is the identity function, the [[equality (mathematics)|equality]] {{math|1=''f''(''x'') = ''x''}} is true for all values of {{mvar|x}} to which {{mvar|f}} can be applied.

==Definition==
Formally, if {{math|''X''}} is a [[Set (mathematics)|set]], the identity function {{math|''f''}} on {{math|''X''}} is defined to be a function with {{math|''X''}} as its [[domain of a function|domain]] and [[codomain]], satisfying
{{bi|left=1.6|{{math|1=''f''(''x'') = ''x''}}   for all elements {{math|''x''}} in {{math|''X''}}.<ref>{{Cite book |last=Knapp |first=Anthony W. |title=Basic algebra |publisher=Springer |year=2006 |isbn=978-0-8176-3248-9}}</ref>}}

In other words, the function value {{math|''f''(''x'')}} in the codomain {{math|''X''}} is always the same as the input element {{math|''x''}} in the domain {{math|''X''}}. The identity function on {{mvar|X}} is clearly an [[injective function]] as well as a [[surjective function]] (its codomain is also its [[range (function)|range]]), so it is [[bijection|bijective]].<ref>{{cite book |last=Mapa |first=Sadhan Kumar |date= 7 April 2014|title=Higher Algebra Abstract and Linear |edition=11th |publisher=Sarat Book House |page=36 |isbn=978-93-80663-24-1}}</ref>

The identity function {{math|''f''}} on {{math|''X''}} is often denoted by {{math|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 {{math|''X''}}.<ref>{{Cite book|url=https://books.google.com/books?id=oIFLAQAAIAAJ&q=the+identity+function+is+given+by+the+identity+relation,+or+diagonal|title=Proceedings of Symposia in Pure Mathematics|date=1974|publisher=American Mathematical Society|isbn=978-0-8218-1425-3|pages=92|language=en|quote=...then the diagonal set determined by M is the identity relation...}}</ref>

==Algebraic properties==
If {{math|''f'' : ''X'' → ''Y''}} is any function, then {{math|1=''f'' ∘ id''X'' = ''f'' = id''Y'' ∘ ''f''}}, where "∘" denotes [[function composition]].<ref>{{cite book
| last = Nel | first = Louis
| year = 2016
| title = Continuity Theory
| url = https://books.google.com/books?id=_JdPDAAAQBAJ&pg=PA21
| page = 21
| publisher = Springer
| location = Cham
| doi = 10.1007/978-3-319-31159-3
| isbn = 978-3-319-31159-3
}}</ref> In particular, {{math|id''X''}} is the [[identity element]] of the [[monoid]] of all functions from {{math|''X''}} to {{math|''X''}} (under function composition).

Since the identity element of a monoid is [[unique (mathematics)|unique]],<ref>{{Cite book|last1=Rosales|first1=J. C.|url=https://books.google.com/books?id=LQsH6m-x8ysC&q=identity+element+of+a+monoid+is+unique&pg=PA1|title=Finitely Generated Commutative Monoids|last2=García-Sánchez|first2=P. A.|date=1999|publisher=Nova Publishers|isbn=978-1-56072-670-8|pages=1|language=en|quote=The element 0 is usually referred to as the identity element and if it exists, it is unique}}</ref> one can alternately define the identity function on {{math|''M''}} to be this identity element. Such a definition generalizes to the concept of an [[identity morphism]] in [[category theory]], where the [[endomorphism]]s of {{math|''M''}} need not be functions.

==Properties==
*The identity function is a [[linear map|linear operator]] when applied to [[vector space]]s.<ref>{{Citation|last=Anton|first=Howard|year=2005|title=Elementary Linear Algebra (Applications Version)|publisher=Wiley International|edition=9th}}</ref>
*In an {{mvar|n}}-[[dimension (vector space)|dimensional]] [[vector space]] the identity function is represented by the [[identity matrix]] {{math|''I''''n''}}, regardless of the [[basis (linear algebra)|basis]] chosen for the space.<ref>{{cite book|title=Applied Linear Algebra and Matrix Analysis|author=T. S. Shores|year=2007|publisher=Springer|isbn=978-038-733-195-9|series=Undergraduate Texts in Mathematics|url=https://books.google.com/books?id=8qwTb9P-iW8C&q=Matrix+Analysis}}</ref>
*The identity function on the positive [[integer]]s is a [[completely multiplicative function]] (essentially multiplication by 1), considered in [[number theory]].<ref>{{cite book|title=Number Theory through Inquiry|author1=D. Marshall |author2=E. Odell |author3=M. Starbird |year=2007|publisher=Mathematical Assn of Amer|isbn=978-0883857519|series=Mathematical Association of America Textbooks}}</ref>
*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 {{math|C1}}).<ref>{{Cite book |last=Anderson |first=James W. |title=Hyperbolic geometry |date=2007 |publisher=Springer |isbn=978-1-85233-934-0 |edition=2. ed., corr. print |series=Springer undergraduate mathematics series |location=London}}</ref>
*In a [[topological space]], the identity function is always [[Continuous_function#Continuous_functions_between_topological_spaces|continuous]].<ref>{{Cite book|last=Conover|first=Robert A.|url=https://books.google.com/books?id=KCziAgAAQBAJ&q=identity+function+is+always+continuous&pg=PA65|title=A First Course in Topology: An Introduction to Mathematical Thinking|date=2014-05-21|publisher=Courier Corporation|isbn=978-0-486-78001-6|pages=65|language=en}}</ref>
*The identity function is [[Idempotence|idempotent]].<ref>{{Cite book|last=Conferences|first=University of Michigan Engineering Summer|url=https://books.google.com/books?id=AvAfAAAAMAAJ&q=The+identity+function+is+idempotent.|title=Foundations of Information Systems Engineering|date=1968|language=en|quote=we see that an identity element of a semigroup is idempotent.}}</ref>

==See also==
* [[Identity matrix]]
* [[Inclusion map]]
* [[Indicator function]]

==References==
{{reflist|30em}}

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Identity function - Revision history

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