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Equip $\mathbb S^n$ with the standard round metric. Let $f : \mathbb S^n \to \mathbb S^n$ be a continous map satisfying $\vert d(f(x),f(y)) - d(x,y)\vert \leq \epsilon$.

Is $f$ is surjective for all $0 \leq \epsilon < \epsilon_0$ for some positive $\epsilon_0$?

My guess would be that the answer is yes and maybe $\epsilon_0 = \pi$.

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Your guess seems to be true. If a map $f$ is not surjective then $f$ can be considered as a continuous map from $S^n$ to $R^n$. Hence there exist two opposite points on $S^n$ which maps to the same point by Borsuk-Ulam theorem.

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  • $\begingroup$ Elegant (but an overkill :-). $\endgroup$ Commented Sep 11, 2014 at 17:36
  • $\begingroup$ @Petya, it was not an overkill, I was just joking. $\endgroup$ Commented Sep 11, 2014 at 18:49

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