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?