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I'm working through "Elementary proof of Borsuk-Ulam Theorem" found here.

Lemma 2: If there exists a continuous mapping of $f: \mathbb S^n \to \mathbb R^n$, which does not identify any pair of antipodes, then there exists an odd continuous mapping $g: \mathbb S^n \to \mathbb S^{n-1}$. Explicit formula for such a mapping $g(x) = \frac{f(x)-f(-x)}{|f(x)-f(-x)|}$.

What does "identify" mean rigorously?

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Seems that it means the following: if $a,b\in\mathbb S^n$ are antipodes then $f(a)\neq f(b)$. – Ilya Aug 10 '11 at 16:23

In this case "to identify" means "to map to the same point". Hence a mapping $f:\mathbb{S}^n\to\mathbb{R}^n$ does not identify any pair of antipodes if and only if $f(x)\neq f(-x)$ for all $x\in\mathbb{S}^n$.

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It means $f(x)=f(-x)$ and that would cause denominator of $g$ being zero.

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