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{{Short description|Type of mathematical function}}
{{distinguish|function constant}}
{{Functions}}
In [[mathematics]], a '''constant function''' is a [[Function (mathematics)|function]] whose (output) value is the same for every input value.
== Basic properties ==
[[Image:wiki constant function 175 200.png|270px|right|thumb|An example of a constant function is {{math|1=''y''(''x'') = 4}}, because the value of {{math|''y''(''x'')}} is 4 regardless of the input value {{mvar|x}}.]]
As a real-valued function of a real-valued argument, a constant function has the general form {{math|1=''y''(''x'') = ''c''}} or just {{nowrap|{{math|1=''y'' = ''c''}}.}} For example, the function {{math|1=''y''(''x'') = 4}} is the specific constant function where the output value is {{math|1=''c'' = 4}}. The [[___domain of a function|___domain of this function]] is the set of all [[real number]]s. The [[Image (mathematics)|image]] of this function is the [[Singleton (mathematics)|singleton]] set {{math|{{mset|4}}}}. The independent variable {{nowrap|1=''x''}} does not appear on the right side of the function expression and so its value is "vacuously substituted"; namely {{math|1=''y''(0) = 4}}, {{math|1=''y''(−2.7) = 4}}, {{math|1=''y''(π) = 4}}, and so on. No matter what value of {{math|''x''}} is input, the output is {{math|4}}.<ref>{{cite book
| last = Tanton | first = James
| year = 2005
| title = Encyclopedia of Mathematics
| publisher = Facts on File, New York
| isbn = 0-8160-5124-0
| page = 94
| url = https://archive.org/details/encyclopedia-of-mathematics_202206/page/94/mode/1up?view=theater
}}</ref>
The graph of the constant function {{math|1=''y'' = ''c''}} is a ''horizontal line'' in the [[plane (geometry)|plane]] that passes through the point {{math|(0, ''c'')}}.<ref>{{cite web|title=College Algebra| last1=Dawkins|first1=Paul| year=2007| publisher= Lamar University|url=http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx| page=224|access-date=January 12, 2014}}</ref> In the context of a [[polynomial]] in one variable {{math|''x''}}, the constant function is called ''non-zero constant function'' because it is a polynomial of degree 0, and its general form is {{math|1=''f''(''x'') = ''c''}}, where {{mvar|c}} is nonzero. This function has no intersection point with the {{nowrap|1={{math|1=''x''}}-}}axis, meaning it has no [[zero of a function|root (zero)]]. On the other hand, the polynomial {{math|1=''f''(''x'') = 0}} is the ''identically zero function''. It is the (trivial) constant function and every {{math|''x''}} is a root. Its graph is the {{nowrap|1={{math|1=''x''}}-}}axis in the plane.<ref>{{cite book|title=Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition|last1=Carter|first1=John A.|last4=Marks|first4=Daniel|last2=Cuevas|first2=Gilbert J.|last3=Holliday|first3=Berchie|last5=McClure|first5=Melissa S. |publisher=Glencoe/McGraw-Hill School Pub Co|year=2005|isbn=978-0078682278|chapter=1|edition=1|page=22}}</ref> Its graph is symmetric with respect to the {{nowrap|1={{math|1=''y''}}-}}axis, and therefore a constant function is an [[Even and odd functions|even function]].<ref>{{cite book
| last = Young | first = Cynthia Y. | authorlink = Cynthia Y. Young
| year = 2021
| title = Precalculus
| edition = 3rd
| url = https://books.google.com/books?id=BOBDEAAAQBAJ&pg=PA122
| page = 122
| publisher = John Wiley & Sons
| isbn = 978-1-119-58294-6 }}</ref>
In
| last1 = Varberg | first1 = Dale E.
| last2 = Purcell | first2 = Edwin J.
| last3 = Rigdon | first3 = Steven E.
| title = Calculus
| year = 2007
| publisher = [[Pearson Prentice Hall]]
| page = 107
| edition = 9th
| isbn = 978-0131469686
}}</ref> This is often written: <math>(x \mapsto c)' = 0</math>. The converse is also true. Namely, if {{math|''y''′(''x'') {{=}} 0}} for all real numbers {{math|''x''}}, then {{math|''y''}} is a constant function.<ref>{{cite web|url=http://www.proofwiki.org/wiki/Zero_Derivative_implies_Constant_Function|title=Zero Derivative implies Constant Function|access-date=January 12, 2014}}</ref> For example, given the constant function {{nowrap|<math>y(x) = -\sqrt{2}</math>.}} The derivative of {{math|''y''}} is the identically zero function {{nowrap|<math>y'(x) = \left(x \mapsto -\sqrt{2}\right)' = 0</math>.}}
== Other properties ==
For functions between [[preorder|preordered sets]], constant functions are both [[order-preserving]] and [[order-reversing]]; conversely, if {{math|''f''}} is both order-preserving and order-reversing, and if the [[Domain of a function|___domain]] of {{math|''f''}} is a [[lattice (order)|lattice]], then {{math|''f''}} must be constant.
* Every constant function whose [[Domain of a function|___domain]] and [[codomain]] are the same set {{math|''X''}} is a [[left zero]] of the [[full transformation monoid]] on {{math|''X''}}, which implies that it is also [[idempotent]].
▲In contexts where it is defined, the [[derivative]] of a function measures how that function varies with respect to the variation of some argument. It follows that, since a constant function does not vary, it's derivative(s), where defined, will be zero. Thus for example:
* It has zero [[slope]] or [[gradient]].
* Every constant function between [[topological space]]s is [[continuous function (topology)|continuous]].▼
* A constant function factors through the [[singleton (mathematics)|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).<ref>{{cite arXiv|last1=Leinster|first1=Tom|title=An informal introduction to topos theory|date=27 Jun 2011|eprint=1012.5647|class=math.CT}}</ref>
* For any non-empty {{math|''X''}}, every set {{math|''Y''}} is [[isomorphic]] to the set of constant functions in <math>X \to Y</math>. For any {{math|''X''}} and each element {{math|''y''}} in {{math|''Y''}}, there is a unique function <math>\tilde{y}: X \to Y</math> such that <math>\tilde{y}(x) = y</math> for all <math>x \in X</math>. Conversely, if a function <math>f: X \to Y</math> satisfies <math>f(x) = f(x')</math> for all <math>x, x' \in X</math>, <math>f</math> is by definition a constant function.
** As a corollary, the one-point set is a [[generator (category theory)|generator]] in the category of sets.
** Every set <math>X</math> is canonically isomorphic to the function set <math>X^1</math>, or [[hom set]] <math>\operatorname{hom}(1,X)</math> 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, <math>\operatorname{hom}(X \times Y, Z) \cong \operatorname{hom}(X(\operatorname{hom}(Y, Z))</math>) 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 <math>\lambda: 1 \times X \cong X \cong X \times 1: \rho</math> [[natural transformation|natural in {{math|''X''}}]], the left and right unitors are the projections <math>p_1</math> and <math>p_2</math> the [[ordered pair]]s <math>(*, x)</math> and <math>(x, *)</math> respectively to the element <math>x</math>, where <math>*</math> is the unique [[point (mathematics)|point]] in the one-point set.
A function on a [[connected set]] is [[locally constant]] if and only if it is constant.
<!--Lfahlberg 01.2014: Perhaps needs information contained in: http://mathworld.wolfram.com/ConstantMap.html, http://www.proofwiki.org/wiki/Definition:Constant_Mapping, http://math.stackexchange.com/questions/133257/show-that-a-constant-mapping-between-metric-spaces-is-continuous and programming http://www.w3schools.com/php/func_misc_constant.asp, http://www2.math.uu.se/research/telecom/software/stcounting.html -->
== References ==▼
{{reflist}}
* Herrlich, Horst and Strecker, George E., ''Category Theory'',
▲* Every constant function between [[topological space]]s is [[continuous]].
== External links ==
▲==References==
{{commons category|Constant functions}}
▲Herrlich, Horst and Strecker, George E., ''Category Theory'', Allen and Bacon, Inc. Boston (1973)
* {{MathWorld |title=Constant Function |id=ConstantFunction}}
* {{planetmath reference |urlname=ConstantFunction |title=Constant function}}
{{polynomials}} {{Functions navbox}}
[[Category:Elementary mathematics]]
[[Category:Elementary special functions]]
[[Category:Polynomial functions]]
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