Solutions to functional equation $f(f(x))=x$ Is there any more solutions to this functional equation $f(f(x))=x$?
I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.

If you don't make any niceness assumptions about $f$, there are lots. Partition $\Bbb{R}$ (or whatever you want $f$'s domain to be) into $1$- and $2$-element subsets, in any way you like. Then define $f(x)=y$, if $\{x, y\}$ is in your partition, or $f(x)=x$, if $\{x\}$ is in your partition.
Moreover, any such $f$ yields such a partition, into the sets $\{x, f(x)\}$. So this is a complete description of all solutions.
Of course, $f$ will be wildly discontinuous for most choices of partition.
