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 $\epsilon_0 = \pi$.