Cardinal invariants concerning functions whose product is almost continuous (Q1890625)

The authors prove that the smallest cardinality of a family \({\mathcal F}\) of real functions for which there is no nonzero function \(g : \mathbb{R} \to \mathbb{R}\) with the property that \(f \cdot g\) is almost continuous (connected function, Darboux function, respectively) for all \(f \in {\mathcal F}\), is equal to the cofinality of the continuum.