Fixed point in a continuous map 
Possible Duplicate:
Periodic orbits 

Suppose that $f$ is a continuous map from $\mathbb R$ to $\mathbb R$, which satisfies $f(f(x)) = x$ for each $x \in \mathbb{R}$.
Does $f$ necessarily have a fixed point?

Yes. This is a direct consequence of Sharkovsky's theorem. And the requirement $f(f(x))=x$ for each $x\in\mathbb{R}$ is too strong. Actually we only need $f(f(x))=x$ for some $x\in\mathbb{R}$ to guarantee that $f$ has a fixed point.
