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>
== Basic properties ==
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''}}.}}<ref>{{Citecite book web|lasttitle=WeissteinEncyclopedia of Mathematics |firstlast1=EricTanton W.|titlefirst1=ConstantJames Function|urlyear=https://mathworld.wolfram.com/ConstantFunction.html2005 |access-datepublisher=2020Facts on File, New York |isbn=0-078160-275124-0 |websitepage=mathworld.wolfram.com|language=en94 }}</ref> For 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}}.
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>
