Are there continuous functions $f:I\to S^2$ such that $f^{-1}(\{x\})$ is infinite for every $x\in S^2$?
Here, $I=[0,1]$ and $S^2$ is the unit sphere.
I have no idea how to do this.
Note: This is not homework! The question came up when I was thinking about something else.
Consider a space filling curve $\gamma: I \rightarrow I^2$, the projection $q: I^2 \rightarrow S^2$ given by the quotient topology on the square that furnishes the sphere, and the projection $\pi: I^2 \rightarrow I$ on the first coordinate.
The map $q \circ \gamma \circ \pi \circ \gamma$ satisfies what you want.