Editing Functional equation (section)

===Involutions===

The [[involution (mathematics)|involution]]s are characterized by the functional equation <math>f(f(x)) = x</math>. These appear in [[Charles Babbage|Babbage's]] functional equation (1820),<ref>{{Cite journal | doi = 10.2307/2007270| jstor = 2007270| title = On Certain Real Solutions of Babbage's Functional Equation| journal = The Annals of Mathematics| volume = 17| issue = 3| pages = 113–122| year = 1916| last1 = Ritt | first1 = J. F. | author1-link = Joseph Ritt}}</ref>
: <math>f(f(x)) = 1-(1-x) = x \, .</math>

Other involutions, and solutions of the equation, include

*<math> f(x) = a-x\, ,</math>
*<math> f(x) = \frac{a}{x}\, ,</math> and 
*<math> f(x) = \frac{b-x}{1+cx} ~ ,</math>
which includes the previous three as [[special case]]s or limits.