{{Short description|Function that returns its argument unchanged}} {{distinguish|Null function|Empty function}} [[image:Function-x.svg|thumb|Graph of the identity function on the real numbers]]
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 <math>f</math> is the identity function, the equality <math>f(x)=x</math> is true for all values of <math>x</math> to which <math>f</math> can be applied.
==Definition== Formally, if <math>X</math> is a set, the identity function <math>f</math> on <math>X</math> is defined to be a function with <math>X</math> as its domain and codomain, satisfying {{block indent | <math>f(x)=x</math> for all elements <math>x</math> in <math>X</math>.<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)</math> in the codomain <math>X</math> is always the same as the input element <math>x</math> in the domain <math>X</math>. The identity function on <math>X</math> is clearly an injective function as well as a surjective function (its codomain is also its range), so it is 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</math> on <math>X</math> is often denoted by <math>\mathrm{id}_X</math>.
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</math>.<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\rightarrow Y</math> is any function, then <math>f\circ\mathrm{id}_X=f=\mathrm{id}_Y\circ f</math>, where "<math>\circ</math>" 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>\mathrm{id}_X</math> is the identity element of the monoid of all functions from <math>X</math> to <math>X</math> (under function composition).
Since the identity element of a monoid is 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</math> to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of <math>M</math> need not be functions.
==Properties== *The identity function is a linear operator when applied to vector spaces.<ref>{{Citation|last=Anton|first=Howard|year=2005|title=Elementary Linear Algebra (Applications Version)|publisher=Wiley International|edition=9th}}</ref> *In an <math>n</math>-dimensional vector space the identity function is represented by the identity matrix <math>I_n</math>, regardless of the 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 integers 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>\mathrm{C}_1</math>).<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.<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 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> *Every map from a set of a single element to itself is necessarily the identity map.
==See also== * Identity matrix * Inclusion map * Indicator function
==References== {{reflist|30em}}
{{Functions navbox}} {{DEFAULTSORT:Identity Function}} Category:Functions and mappings Category:Elementary mathematics Category:Basic concepts in set theory Category:Types of functions Category:1 (number)