How to show $E:=\{x\in [0,1]: |f^{-1}(\{x\})|\,\mbox{ is even} \}$ is countable?

real-analysis

Hi everybody
I have seen the following question which I could not solve it, so I thought I can share the question with you and ask for help.

Question: Let $f:[0,1]\to [0,1]$ be a continuous function such that $f(0)=0$ and $f(1)=1$. Moreover assume $f^{-1}(\{x\})$ is finite for all $x$. Prove $$E:=\{x\in [0,1]: |f^{-1}(\{x\})|\,\mbox{ is even} \}$$
is countable.


Building on my comment: For a local maximum $y$ with function value $f(y)$, let $\rho(y)$ be the radius within which all function values are less than $f(y)$. Let $\rho_n=1/n$, and consider the local maxima with $\rho_n\ge\rho(y)>\rho_{n+1}$. There can't be more than $n$ of these, since their radii can't overlap. So we can enumerate all local maxima (and analogously all local minima).

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