Editing Fixed point (mathematics) (section)

== Fixed point of a function ==

Formally, {{mvar|c}} is a fixed point of a function {{mvar|f}} if {{mvar|c}} belongs to both the [[domain of a function|domain]] and the [[codomain]] of {{mvar|f}}, and {{math|1=''f''(''c'') = ''c''}}.
In particular, {{mvar|f}} cannot have any fixed point if its domain is disjoint from its codomain.
If {{math|''f''}} is defined on the [[real number]]s, it corresponds, in graphical terms, to a [[curve]] in the [[Euclidean plane]], and each fixed-point {{math|''c''}} corresponds to an intersection of the curve with the line {{math|1=''y''&thinsp;=&thinsp;''x''}}, cf. picture.

For example, if {{math|''f''}} is defined on the [[real number]]s by
<math> f(x) = x^2 - 3 x + 4,</math>
then 2 is a fixed point of {{math|''f''}}, because  {{math|1=''f''(2) = 2}}.

Not all functions have fixed points: for example, {{math|1=''f''(''x'') = ''x'' + 1}} has no fixed points because {{math|''x'' + 1}} is never equal to {{math|''x''}} for any real number.