Brouwer fixed point theorem

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• Euler and Lefschetz numbers#Fixed points

Theorem. Any continuous function on a closed unit ball has a fixed point.

In other words, suppose ${\bf B}$ $= \{ x \in {\bf R}^n: ||x|| \le 1 \} $ is the closed unit ball in ${\bf R}^n$ and $f: {\bf B}\to {\bf B}$ is a continuous function. Then there is $x\in {\bf B}$ such that $f(x)=x$.

The theorem holds if the ball is replaced with anything homeomorphic to it.

It is a particular case of the Lefschetz fixed point theorem.