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constant 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 basic properties 2 other properties 3 references 4 external links toggle the table of contents constant function 43 languages العربية bosanski català čeština чӑвашла dansk deutsch ελληνικά esperanto español euskara فارسی suomi français galego עברית magyar bahasa indonesia ido íslenska italiano 日本語 한국어 latina македонски नेपाली polski português română русский srpskohrvatski српскохрватски simple english slovenčina српски srpski svenska தமிழ் ไทย türkçe українська tiếng việt 閩南語 bân lâm gí 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 wikimedia commons wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia type of mathematical function not to be confused with function constant function x f x history of the function concept types by domain and codomain x 𝔹 𝔹 x 𝔹 n x x ℤ ℤ x x ℝ ℝ x ℝ n x x ℂ ℂ x ℂ n 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 list of specific functions v t e in mathematics a constant function is a function whose output value is the same for every input value basic properties edit an example of a constant function is y x 4 because the value of y x is 4 regardless of the input value x as a real valued function of a real valued argument a constant function has the general form y x c or just y c for example the function y x 4 is the specific constant function where the output value is c 4 the domain of this function is the set of all real numbers the image of this function is the singleton set 4 the independent variable x does not appear on the right side of the function expression and so its value is vacuously substituted namely y 0 4 y 2 7 4 y π 4 and so on no matter what value of x is input the output is 4 1 the graph of the constant function y c is a horizontal line in the plane that passes through the point 0 c 2 in the context of a polynomial in one variable x the constant function is called non zero constant function because it is a polynomial of degree 0 and its general form is f x c where c is nonzero this function has no intersection point with the x axis meaning it has no root zero on the other hand the polynomial f x 0 is the identically zero function it is the trivial constant function and every x is a root its graph is the x axis in the plane 3 its graph is symmetric with respect to the y axis and therefore a constant function is an even function 4 in the context where it is defined the derivative of a function is a measure of the rate of change of function values with respect to change in input values because a constant function does not change its derivative is 0 5 this is often written x c 0 displaystyle x mapsto c 0 the converse is also true namely if y x 0 for all real numbers x then y is a constant function 6 for example given the constant function y x 2 displaystyle y x sqrt 2 the derivative of y is the identically zero function y x x 2 0 displaystyle y x left x mapsto sqrt 2 right 0 other properties edit for functions between preordered sets constant functions are both order preserving and order reversing conversely if f is both order preserving and order reversing and if the domain of f is a lattice then f must be constant every constant function whose domain and codomain are the same set x is a left zero of the full transformation monoid on x which implies that it is also idempotent it has zero slope or gradient every constant function between topological spaces is continuous a constant function factors through the one point set the terminal object in the category of sets this observation is instrumental for f william lawvere s axiomatization of set theory the elementary theory of the category of sets etcs 7 for any non empty x every set y is isomorphic to the set of constant functions in x y displaystyle x to y for any x and each element y in y there is a unique function y x y displaystyle tilde y x to y such that y x y displaystyle tilde y x y for all x x displaystyle x in x conversely if a function f x y displaystyle f x to y satisfies f x f x displaystyle f x f x for all x x x displaystyle x x in x f displaystyle f is by definition a constant function as a corollary the one point set is a generator in the category of sets every set x displaystyle x is canonically isomorphic to the function set x 1 displaystyle x 1 or hom set hom 1 x displaystyle operatorname hom 1 x in the category of sets where 1 is the one point set because of this and the adjunction between cartesian products and hom in the category of sets so there is a canonical isomorphism between functions of two variables and functions of one variable valued in functions of another single variable hom x y z hom x hom y z displaystyle operatorname hom x times y z cong operatorname hom x operatorname hom y z the category of sets is a closed monoidal category with the cartesian product of sets as tensor product and the one point set as tensor unit in the isomorphisms λ 1 x x x 1 ρ displaystyle lambda 1 times x cong x cong x times 1 rho natural in x the left and right unitors are the projections p 1 displaystyle p_ 1 and p 2 displaystyle p_ 2 the ordered pairs x displaystyle x and x displaystyle x respectively to the element x displaystyle x where displaystyle is the unique point in the one point set a function on a connected set is locally constant if and only if it is constant references edit tanton james 2005 encyclopedia of mathematics facts on file new york p 94 isbn 0 8160 5124 0 dawkins paul 2007 college algebra lamar university p 224 retrieved january 12 2014 carter john a cuevas gilbert j holliday berchie marks daniel mcclure melissa s 2005 1 advanced mathematical concepts pre calculus with applications student edition 1 ed glencoe mcgraw hill school pub co p 22 isbn 978 0078682278 young cynthia y 2021 precalculus 3rd ed john wiley sons p 122 isbn 978 1 119 58294 6 varberg dale e purcell edwin j rigdon steven e 2007 calculus 9th ed pearson prentice hall p 107 isbn 978 0131469686 zero derivative implies constant function retrieved january 12 2014 leinster tom 27 jun 2011 an informal introduction to topos theory arxiv 1012 5647 math ct herrlich horst and strecker george e category theory heldermann verlag 2007 external links edit wikimedia commons has media related to constant functions weisstein eric w constant function mathworld constant function planetmath v t e polynomials and polynomial functions and polynomial equations by degree zero polynomial degree undefined or 1 or constant function 0 linear function 1 linear equation quadratic function 2 quadratic equation cubic function 3 cubic equation quartic function 4 quartic equation quintic function 5 sextic equation 6 septic equation 7 by properties univariate bivariate multivariate monomial binomial trinomial irreducible square free homogeneous quasi homogeneous tools and algorithms factorization greatest common divisor division horner s method of evaluation polynomial identity testing resultant discriminant gröbner basis 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 constant_function oldid 1350437448 categories elementary mathematics elementary special functions polynomial functions hidden categories articles with short description short description is different from wikidata commons category link is on wikidata this page was last edited on 21 april 2026 at 23 30 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 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