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identity function wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 definition 2 algebraic properties 3 properties 4 see also 5 references toggle the table of contents identity function 42 languages العربية bosanski català corsu čeština чӑвашла dansk deutsch ελληνικά esperanto español eesti euskara فارسی suomi français galego עברית हिन्दी hrvatski bahasa indonesia íslenska italiano 日本語 한국어 lombard नेपाली nederlands polski português română русский simple english slovenščina svenska தமிழ் ไทย türkçe українська tiếng việt 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia function that returns its argument unchanged not to be confused with null function or empty function 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 f displaystyle f is the identity function the equality 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 edit formally if x displaystyle x is a set the identity function f displaystyle f on x displaystyle x is defined to be a function with x displaystyle x as its domain and 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 as well as a surjective function its codomain is also its range so it is bijective 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 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 displaystyle x 3 algebraic properties edit 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 4 in particular i d x displaystyle mathrm id _ x is the identity element of the monoid of all functions from x displaystyle x to x displaystyle x under function composition since the identity element of a monoid is unique 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 in category theory where the endomorphisms of m displaystyle m need not be functions properties edit the identity function is a linear operator when applied to vector spaces 6 in an n displaystyle n dimensional vector space the identity function is represented by the identity matrix i n displaystyle i_ n regardless of the basis chosen for the space 7 the identity function on the positive integers is a completely multiplicative function essentially multiplication by 1 considered in number theory 8 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 c 1 displaystyle mathrm c _ 1 9 in a topological space the identity function is always continuous 10 the identity function is idempotent 11 every map from a set of a single element to itself is necessarily the identity map see also edit identity matrix inclusion map indicator function references edit knapp anthony w 2006 basic algebra springer isbn 978 0 8176 3248 9 mapa sadhan kumar 7 april 2014 higher algebra abstract and linear 11th ed sarat book house p 36 isbn 978 93 80663 24 1 proceedings of symposia in pure mathematics american mathematical society 1974 p 92 isbn 978 0 8218 1425 3 then the diagonal set determined by m is the identity relation nel louis 2016 continuity theory cham springer p 21 doi 10 1007 978 3 319 31159 3 isbn 978 3 319 31159 3 rosales j c garcía sánchez p a 1999 finitely generated commutative monoids nova publishers p 1 isbn 978 1 56072 670 8 the element 0 is usually referred to as the identity element and if it exists it is unique anton howard 2005 elementary linear algebra applications version 9th ed wiley international t s shores 2007 applied linear algebra and matrix analysis undergraduate texts in mathematics springer isbn 978 038 733 195 9 d marshall e odell m starbird 2007 number theory through inquiry mathematical association of america textbooks mathematical assn of amer isbn 978 0883857519 anderson james w 2007 hyperbolic geometry springer undergraduate mathematics series 2 ed corr print ed london springer isbn 978 1 85233 934 0 conover robert a 2014 05 21 a first course in topology an introduction to mathematical thinking courier corporation p 65 isbn 978 0 486 78001 6 conferences university of michigan engineering summer 1968 foundations of information systems engineering 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 retrieved from https en wikipedia org w index php title identity_function oldid 1341292889 categories functions and mappings elementary mathematics basic concepts in set theory types of functions 1 number hidden categories articles with short description short description is different from wikidata this page was last edited on 2 march 2026 at 10 50 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct 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