Brouwer's Fixed Point Theorem

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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.


Smooth Mapping

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

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


Source of Name

This entry was named for Luitzen Egbertus Jan Brouwer.