Darboux like functions that are characterizable by images, preimages and associated sets (Q1978880)

The class \(\mathcal F\) of functions is characterizable by images (preimages, respectively), if there exist two systems \(\mathcal A\) and \(\mathcal B\) of subsets of \(\mathbb R\) such that \[ \begin{aligned} \mathcal F &= \{f:\mathbb R \to \mathbb R;\;\forall A \in \mathcal A:\;f(A) \in \mathcal B\} \\ (\mathcal F &= \{f:\mathbb R \to \mathbb R;\;\forall B \in \mathcal B:\^^Mf^{-1}(B) \in \mathcal A\}, \text{respectively}). \end{aligned} \] They show which classes of Darboux-like functions (e.g., almost continuous functions, connectivity functions, etc.) are characterizable by images (preimages, respectively) and which are not. It is also shown that the class of Sierpiński-Zygmund functions can be characterized neither by images nor preimages. (A function \(f\) is a Sierpiński-Zygmund function if a restriction \(f|Y\) is discontinuous whenever \(Y \subset \mathbb R\) is a set of cardinality continuum).