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identity function wikipedia 238 captures 15 jan 2004 31 aug 2026 sep oct nov 08 2021 2022 2023 success fail about this capture timestamps the wayback machine http web archive org web 20221008213534 http en wikipedia org wiki identity_function identity function from wikipedia the free encyclopedia jump to navigation jump to search in mathematics a function that always returns the same value that was used as its argument 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 is the identity function the equality f x x is true for all values of x to which f can be applied contents 1 definition 2 algebraic properties 3 properties 4 see also 5 references definition edit formally if m is a set the identity function f on m is defined to be a function with m as its domain and codomain satisfying f x x for all elements x in m 1 in other words the function value f x in the codomain m is always the same as the input element x in the domain m the identity function on m is clearly an injective function as well as a surjective function so it is bijective 2 the identity function f on m is often denoted by id m 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 m 3 algebraic properties edit if f m n is any function then we have f id m f id n f where denotes function composition in particular id m is the identity element of the monoid of all functions from m to m under function composition since the identity element of a monoid is unique 4 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 endomorphisms of m need not be functions properties edit the identity function is a linear operator when applied to vector spaces 5 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 6 the identity function on the positive integers is a completely multiplicative function essentially multiplication by 1 considered in number theory 7 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 8 in a topological space the identity function is always continuous 9 the identity function is idempotent 10 see also edit identity matrix inclusion map 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 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 james w anderson hyperbolic geometry springer 2005 isbn 1 85233 934 9 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 retrieved from https en wikipedia org w index php title identity_function oldid 1076989097 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 navigation menu personal tools not logged in talk contributions create account log in namespaces article talk english views read edit view history more search navigation main page contents current events random article about wikipedia contact us donate contribute help learn to edit community portal recent changes upload file tools what links here related changes upload file special pages permanent link page information cite this page wikidata item print export download as pdf printable version languages العربية bosanski català чӑвашла čeština corsu dansk deutsch eesti ελληνικά español esperanto euskara فارسی français galego 한국어 हिन्दी hrvatski bahasa indonesia íslenska italiano עברית lombard nederlands नेपाली 日本語 polski português română русский slovenščina suomi svenska தமிழ் ไทย türkçe українська tiếng việt 粵語 中文 edit links this page was last edited on 14 march 2022 at 01 17 utc text is available under the creative commons attribution sharealike license 3 0 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 mobile view developers statistics cookie statement