Borsuk-Ulam theorem

This article describes a theorem about spheres

Statement

Let $S^{m}$ denote the $m$-dimensional sphere (embedded in $\mathbb {R} ^{m+1}$) and $\mathbb {R} ^{m}$ denote $m$-dimensional Euclidean space. Then, given any continuous map $f:S^{m}\to \mathbb {R} ^{m}$, there is a point $x\in S^{m}$ such that $f(x)=f(-x)$.