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

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".

• $f(x)=c-x$ is also a solution. Sep 27, 2015 at 15:04
• @AhmedS.Attaalla It's not - point $(0,1)$ belongs to this line, but $(1,0)$ doesn't. Sep 27, 2015 at 15:07
• Sep 27, 2015 at 15:08
• Wow how did I miss that. I was thinking parrellel Sep 27, 2015 at 15:08