Associated sets of \(Baire^*\) 1 functions (Q795943)

Let \(I\subset R\) be an interval, f:\(I\to R\), \(E^{\alpha}(f)=\{x\in I:f(x)<\alpha \}\) and \(E_{\alpha}(f)=\{x\in I:f(x)>\alpha \}\) for \(\alpha \in R\), \({\mathcal B}^*_ 1\) denote the class of all f such that, for every perfect set \(P\subset I\), there is an open intervall \(J\subset I\) such that \(P\cap J\neq \emptyset\) and \(f| P\cap J\) is continuous. By using results of \textit{M. Laczkovich} and the reviewer [Stud. Sci. Math. Hungar. 10, 463-472 (1975; Zbl 0405.26006); Acta Math. Acad. Sci. Hung. 33, 51-70 (1979; Zbl 0401.54010)] the authors characterize the sets \(E^{\alpha}(f)\) and \(E_{\alpha}(f)\) for \(f\in {\mathcal B}^*_ 1\) and for \(f\in {\mathcal D}\cap {\mathcal B}^*_ 1\) where \({\mathcal D}\) denotes the class of Darboux functions.