Find continuous functions that satisfy $f(f(x))=x$ over the reals.

Question

I'm looking for a method to solve:

$$f(f(x))=x$$

Where $f$ is defined for $x \in R$

So far by inverting both sides I have:

$f(x)=f^{-1}(x)$

Which means that my function should be symmetrical over $y=x$. I may "guess" the functions:

$y=x$

$y=c-x$

However I'm wondering is there a way to solve this without "guessing".

@AhmedS.Attaalla It's not - point $(0,1)$ belongs to this line, but $(1,0)$ doesn't. — Wojowu, Sep 27 '15 at 15:07
Related: http://math.stackexchange.com/questions/46635/examples-of-involutions-on-mathbbr — MonadBoy, Sep 27 '15 at 15:08

Joseph Zambrano · Accepted Answer · 2015-09-27T15:10:59.003

6

Functions satisfying this property are known as involutions. See the article for many examples in various fields. You are correct about the symmetry condition.

Edit: I should add that there are unaccountably infinitely many of such functions (even restricting to continuous functions).

edited Sep 27 '15 at 15:10

answered Sep 27 '15 at 15:05

Joseph Zambrano

1,237

Find continuous functions that satisfy $f(f(x))=x$ over the reals.

1 Answers1