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Number of even images is very small

I found on Math Stack Exchange a generalisation of a problem I proposed in the Romanian Mathematical Olympiad:

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

My problem was to prove that there doesn’t exist a function $f:\Bbb{R} \to \Bbb{R}$ with the property that the equation $f(x)=y$ has an even number of solutions for all $y \in \Bbb{R}$.