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{{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 ==
| 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
▲: '''Example:''' The function {{math|1=''y''(''x'') = 2}} or just {{math|1=''y'' = 2}} is the specific constant function where the output value is {{math|1=''c'' = 2}}. The [[___domain of a function|___domain of this function]] is the set of all real numbers '''R'''. The [[Image (mathematics)|image]] of this function is the [[Singleton (mathematics)|singleton]] set {{math|{{mset|2}}}}. The independent variable ''x'' does not appear on the right side of the function expression and so its value is "vacuously substituted". Namely {{math|1=''y''(0) = 2}}, {{math|1=''y''(−2.7) = 2}}, {{math|1=''y''(π) = 2}}, and so on. No matter what value of {{math|''x''}} is input, the output is {{math|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
▲In the context of a [[polynomial]] in one variable {{math|''x''}}, the '''non-zero constant function''' 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 ''x''-axis, that is, 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 ''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>
| 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
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| 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 ==
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* 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.
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* {{planetmath reference |urlname=ConstantFunction |title=Constant function}}
{{polynomials}} {{Functions navbox}}
[[Category:Elementary mathematics]]
[[Category:Elementary special functions]]
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