IMC 2007 Problem 9

Let $C$ be a nonempty closed bounded subset of the real line and $f: C\to C$ be a nondecreasing continuous function. Show that there exists a point $p\in C$ such that $f(p) = p$ . (A set is closed if its complement is a union of open intervals. A function $g$ is nondecreasing if $g(x)\le g(y)$ for all $x\le y$ .)