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".

Answer 1

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).