Continuous functions taking every value a given number of times

TL;DR

This paper characterizes when a function from [0,1] to a set of values can be realized as the preimage size of a continuous function, providing necessary and sufficient conditions.

Contribution

It offers a complete characterization of functions for which a continuous function with prescribed preimage cardinalities exists.

Findings

Provides necessary and sufficient conditions for the existence of such continuous functions.

Establishes a link between the target value set and the structure of preimages.

Advances understanding of inverse image structures in continuous functions.

Abstract

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