# Brouwer's Fixed Point Theorem/General Case

## Theorem

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

$\forall f \in \map {C^0} { {B^n}^- \to {B^n}^- } : \exists x \in {B^n}^- : \map f x = x$