Continuous functions taking every value a given number of times

Abstract

We give necessary and sufficient conditions on a function $f:[0,1]\to {0,1,2,...,ω,\continuum}$ under which there exists a continuous function $F:[0,1]\to [0,1]$ such that for every $y\in [0,1]$ we have $|F^{-1}(y)|=f(y)$.

16 pages, 2 figures