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. here is an outline of how to show this:
$\ \ \ \ $1) Show that the range of $f$ is $\Bbb R$.
$\ \ \ \ $2) Show that $f$ is one-to-one; hence, monotone by continuity.
Assume $f$ is not the identity function (otherwise the result is trivial).
Then:
$\ \ \ \ $3) Show that $f$ is strictly decreasing.
Then, using 1) and 3): 
$\ \ \ \ $4) Show that the function $g$ defined by $g(x)=f(x)-x$ has a (unique) zero. 
Conclude that $f$ has a (unique) fixed point.
