Continuous functions taking every value a given number of times (Q1046828)

Conditions characterizing the indicatrices \(y\in [0,1]\mapsto | f^{-1}(y)| \) of continuous functions \(f:[0,1]\to [0,1]\) are found. While it is known to be a necessary condition that the set where the indicatrix is uncountable is analytic, and the other conditions describing how the indicatrix behaves locally have a short proof, the sufficiency of these conditions uses a quite involved construction.