Home > Analysis > Number of even images is very small

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}.

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