In the pages 135-136 of Algebraic Topology, Hatcher studies the degree of a map $f:S^n\to S^n$ in terms of the local degree at the preimages of a point $y\in S^n$ such that $f^{-1}(y)$ is a finite set. I wonder if there is for some $n>0$ an example of a map $f:S^n\to S^n$ such that $f^{-1}(y)$ is infinite for all $y$.
Every map I've thought so far has points with finite preimage. In additiom it can be shown that this map can't be differentiable. Therefore it must be a weird map. Does any one know one such example o at least what other properties should it verify?
Here's an example for $n=1$.
Consider a space-filling map $f:[0,1]\to[0,1]\times[0,1]$, that is a continuous surjection. Consider $f_1$, the first component of it. This is a continuous map $[0,1]\to[0,1]$ whose fibres are all uncountable. If we compose with a surjection $[0,1]$ to $S^1$ we get a continuous $g: [0,1]\to S^1$ with all fibres uncountable. Now define $h:S^1\to S^1$ by $h(x,y)=g(|x|)$; then $h$ is a continuous surjection with all fibres uncountable.
One can use similar tricks for any $n$.