TITLE: If $f^3=\rm id$ then it is identity function
QUESTION [7 upvotes]: Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $f^3(x)=f\circ f\circ f(x)=x$ for all $x$.
How can I prove $f$ is the identity function?

REPLY [3 votes]: First show that $f^{-1}$ is strictly increasing. The comments above give hints how to show this. (You note that $f^{-1}=f^2$, hence $f$ is strictly monotone by continuity, hence $f^{-1}$ is strictly increasing). Since $f^{-1}$ is strictly increasing it follows that $f$ is strictly increasing.  
Now assume that $f(x)>x$ some $x \in \mathbb{R}$, then $f^2(x)>f(x)$, implying that $f^3(x)>f^2(x)$, hence $f^3(x)>f^2(x)>f(x)>x$. A contradiction.