Borsuk-Ulam theorem

In mathematics, the Borsuk–Ulam theorem (BUT), states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.

Formally: if ${\displaystyle f:S^{n}\to \mathbb {R} ^{n}}$ is continuous then there exists an ${\displaystyle x\in S^{n}}$ such that: ${\displaystyle f(-x)=f(x)}$.

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