Brouwer's Fixed Point Theorem

Theorem

One-Dimensional Version

Let $f: \left[{a .. b}\right] \to \left[{a .. b}\right]$ be a real function which is continuous on the closed interval $\left[{a .. b}\right]$.

Then:

$\exists \xi \in \left[{a .. b}\right]: f \left({\xi}\right) = \xi$.

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

General Case

Any smooth map $f$ of the closed unit ball $B^n \subset \R^n$ into itself must have a fixed point:

$\forall f \in C^\infty (B^n \to B^n): \exists x \in B^n: f(x)=x$

Source of Name

This entry was named for Luitzen Egbertus Jan Brouwer.