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.