When the degree of a map $S^n \rightarrow S^n$ isn't 1 or -1, are there always two points that map to the same point and whose distance from each other can be bounded below by a function of the dimension and degree alone?
In other words, for every $n \in \mathbb{N}$ and $k \in \mathbb{Z} \backslash \{1,-1\}$ does there exist $\epsilon > 0$ such that for every continuous $f:S^n \rightarrow S^n$ with degree $k$, there must exist $x,y$ such that $|x-y|\ge\epsilon$ and $f(x)=f(y)$?
The case when $n=1$ is easy enough. Is there a similarly simple argument for higher dimensions?
This question arose when I wondered whether an injective continuous map $f:S^n \rightarrow S^n$ must have degree 1 or -1, which I was surprised to not be able to Google up an answer to right away.