Editing Constant function (section)

== 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 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
 | 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>.}}