# Brouwer's Fixed Point Theorem

## Theorem

### One-Dimensional Version

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.

### Smooth Mapping

A smooth mapping $f$ of the closed unit ball $B^n \subset \R^n$ into itself has a fixed point:

$\forall f \in \map {C^\infty} {B^n \to B^n}: \exists x \in B^n: \map f x = x$

## Source of Name

This entry was named for Luitzen Egbertus Jan Brouwer.