\(\kappa \)-to-\(1\) Darboux-like functions (Q1978898)

The authors study the following problem: For which functions \(j\) from \(\mathbb{R}\) to cardinals does there exist a Darboux-like function \(f\) such that \[ j(y) = {\text{card}} f^{-1}(y)? \] By a Darboux-like function the authors mean continuous functions, Darboux functions, perfect road functions, functions with the Cantor intermediate value property and functions with the strong Cantor intermediate value property. The sets \(\mathbb{R}^n\), \([0,1]^n, n\in \mathbb{N}\), are considered as domains of these functions. Among other results they answer this question completely when \(j\) attains finite values only and \(f\) is required to be continuous. The characterization of \(j\) is given in terms of level sets. The authors proved also that for every \(j: \mathbb{R} \to \langle 1, 2^{\aleph _0}\rangle \) there exists a desired function with the perfect road property (with the Cantor intermediate value property, respectively).