Sums of continuous and Darboux functions (Q1898992)

Let \({\mathcal C}+ D\) denote the class of real functions which are the sum of a continuous and a Darboux function. Given an interval \(I\) and a set \(A\subset \mathbb{R}\) let \({\mathcal D}^*(I, A)\) denote the set of all \(f: I\to A\) such that all level sets \(f^{- 1}(y)\), \(y\in A\), are dense in \(I\). Assuming that the additivity of the ideal of meager subsets of reals is equal to \(2^\omega\), \textit{B. Kirchheim} and \textit{T. Natkaniec} proved that \({\mathcal D}^*(\mathbb{R}, \mathbb{R}\backslash Q)\backslash({\mathcal C}+ D)\neq \emptyset\) [Real Anal. Exch. 16, No. 2, 481-486 (1991; Zbl 0742.26010)]. The author generalizes this result and proves that the only sets \(A\subset \mathbb{R}\) for which \({\mathcal D}^*(I, A)\subset {\mathcal C}+ D\) are intervals. (He uses the continuum hypothesis in the proof, but this assumption can easily be weakened).