Products of Darboux functions (Q1803978)

In this paper, there are three theorems about the existence of a function \(g: I\to\mathbb{R}\) for a given set \(F\) of real functions defined on the closed interval \(I\) such that \(fg\) is Darboux for any \(f\in F\). Theorem 1. For any countable set \(F\) of Baire functions of the class \(B_ \alpha\), where \(\alpha>1\), there exists a non-zero Darboux function \(g\) in \(B_ \alpha\) such that: (1) \(fg\) is Darboux for any \(f\in F\) and (2) the set \(\{x\in I: g(x)\neq 0\}\) is of the first Baire category in \(\mathbb{R}\) and of Lebesgue measure zero. Theorems 2 and 3 are proved under the hypothesis that the following set- theoretical assertion is true: Any union of less than \(2^ \omega\) sets of the first category in \(\mathbb{R}\) is also of the first category in \(\mathbb{R}\).