Brouwer's Fixed Point Theorem/One-Dimensional Version

Theorem
Let $f: \closedint a b \to \closedint a b$ be a real function which is continuous on the closed interval $\closedint a b$.

Then:
 * $\exists \xi \in \closedint a b: \map f \xi = \xi$

That is, a continuous real function from a closed real interval to itself fixes some point of that interval.