Brouwer fixed-point theorem

This article describes a theorem about spheres

Statement

In the language of spheres

Any continuous map from a disc to itself must have a fixed point. In other words, if $D^n$ denotes the spherical disc in Failed to parse (syntax error): {\displaystyle \R&n} , any continuous map $f:D^n\to D^n$ must have a point $x$ such that $f(x)=x$.

In the language of simplices

Any continuous map from thethe standard $n$-simplex, to itself, has a fixed point.