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{{Short description|Type of mathematical function}}

{{distinguish|function constant}}

{{Functions}}

[[Image:wiki constant function 175 200.png|270px|right|Constant function ''y''=4]]

In [[mathematics]], a '''constant function''' is a [[Function (mathematics)|function]] whose (output) value is the same for every input value.

In [[mathematics]], a '''constant function''' is a [[Function (mathematics)|function]] whose (output) value is the same for every input value.<ref>{{cite book|title=Encyclopedia of Mathematics|last1=Tanton|first1=James|year=2005|publisher=Facts on File, New York|isbn=0-8160-5124-0|page=94}}</ref><ref>{{cite web | url=http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf |title=Oxford Concise Dictionary of Mathematics, Constant Function | author=C.Clapham, J.Nicholson | publisher =Addison-Wesley | year =2009|page=175|accessdate=January 12, 2014}}</ref><ref>{{cite book|title=CRC Concise Encyclopedia of Mathematics|last1=Weisstein|first1=Eric|publisher=CRC Press, London|isbn=0-8493-9640-9|year=1999|page=313}}</ref> For example, the function <math>y(x) = 4</math> is a constant function because the value of &nbsp;<math>y(x)</math> &nbsp;is 4 regardless of the input value <math>x</math> (see image).

== Basic properties ==

As[[Image:wiki a real-valuedconstant function of175 a200.png|270px|right|thumb|An real-valuedexample argument,of a constant function hasis {{math|1=''y''(''x'') = 4}}, because the generalvalue formof &nbsp;<{{math>|''y''(''x'')=c</math>}} &nbsp;oris just4 &nbsp;<math>y=c</math>regardless &nbsp;of the input value {{mvar|x}}.]]

:'''Example:'''As Thea real-valued function &nbsp;<of a real-valued argument, a constant function has the general form {{math>|1=''y''(''x'') =2</math> &nbsp;''c''}} or just &nbsp;<{{nowrap|{{math>|1=''y'' =2</ ''c''}}.}} For example, the function {{math>|1=''y''(''x'') = 4}} &nbsp;is the specific constant function where the output value is &nbsp;<{{math>|1=''c'' =2</math> 4}}. The [[___domain of a function|___domain of this function]] is the set of all [[real numbers ℝnumber]]s. The [[codomainImage (mathematics)|image]] of this function is justthe [[Singleton (mathematics)|singleton]] set {2{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".; Namelynamely {{math|1=''y''(0) =2 4}}, {{math|1=''y''(−2.7) =2 4}}, {{math|1=''y''(π) =2 4}},... and so on. No matter what value of {{math|''x''}} is input, the output is "2"{{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}} \neqis 0</math>&nbsp;nonzero. This function has no intersection point with the {{nowrap|1={{math|1=''x''}}-}}axis, that is,meaning it has no [[Zerozero of a function|root (zero)]]. On the other hand, the polynomial &nbsp;<{{math>|1=''f''(''x'') = 0</math> }} &nbsp;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

:'''Example:''' The function &nbsp;<math>y(x)=2</math> &nbsp;or just &nbsp;<math>y=2</math> &nbsp;is the specific constant function where the output value is &nbsp;<math>c=2</math>. The [[___domain of a function|___domain of this function]] is the set of all real numbers ℝ. The [[codomain]] of this function is just {2}. 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)=2, ''y''(−2.7)=2, ''y''(π)=2,.... No matter what value of ''x'' is input, the output is "2".

| 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 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.<ref>{{cite book

:'''Real-world example:''' A store where every item is sold for the price of 1 euro.

| 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 ==

The graph of the constant function <math>y=c</math> is a '''horizontal line''' in the [[plane (geometry)|plane]] that passes through the point <math>(0,c)</math>.<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|accessdate=January 12, 2014}}</ref>

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 the context of a [[polynomial]] in one variable ''x'', the '''non-zero constant function''' is a polynomial of degree 0 and its general form is <math>f(x) = c \, ,\,\, c \neq 0</math>&nbsp;. This function has no intersection point with the ''x''-axis, that is, it has no [[Zero of a function|root (zero)]]. On the other hand, the polynomial &nbsp;<math>f(x)=0</math> &nbsp;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.<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>

* It has zero [[slope]] or [[gradient]].

A constant function is an [[Even and odd functions|even function]], i.e. the graph of a constant function is symmetric with respect to the ''y''-axis.

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.<ref>{{cite web|url=http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx|title=Derivative Proofs|year=2007|last1=Dawkins|first1=Paul|publisher= Lamar University|accessdate=January 12, 2014}}</ref> This is often written: &nbsp;<math>(c)'=0</math>&nbsp;. The converse is also true. Namely, if ''y''&#39;(''x'')=0 for all real numbers ''x'', then ''y''(''x'') is a constant function.<ref>{{cite web|url=http://www.proofwiki.org/wiki/Zero_Derivative_implies_Constant_Function|title=Zero Derivative implies Constant Function|accessdate=January 12, 2014}}</ref>

:'''Example:''' Given the constant function &nbsp; <math>y(x)=-\sqrt{2}</math> &nbsp;. The derivative of ''y'' is the identically zero function &nbsp; <math>y'(x)=(-\sqrt{2})'=0</math> &nbsp;.

==Other properties==

For functions between [[preorder|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 a function|___domain]] of ''f'' is a [[lattice (order)|lattice]], then ''f'' must be constant.

* Every constant function whose [[Domain of a function|___domain]] and [[codomain]] are the same is [[idempotent]].

* 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>

* EveryFor setany non-empty {{math|''X''}}, every set {{math|''Y''}} is [[isomorphic]] to the set of constant functions intoin it.<math>X \to Y</math>. For eachany element x{{math|''X''}} and anyeach setelement {{math|''y''}} in {{math|''Y''}}, there is a unique function <math>\tilde{xy}: YX \rightarrowto XY</math> such that <math>\tilde{xy}(yx) = xy</math> for all <math>yx \in YX</math>. Conversely, if a function <math>f: YX \rightarrowto XY</math> satisfies <math>f(yx) = f(yx')</math> for all <math>yx, yx' \in YX</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 cartesianCartesian 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 [[cartesianCartesian 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 \times1times 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'', Heldermann Verlag (2007).

== External links ==

{{commons category|Constant functions}}

* {{MathWorld |title=Constant Function |id=ConstantFunction}}

* {{planetmath reference |idurlname=4727ConstantFunction |title=Constant function}}

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