Products of Baire one double star functions
The symbol \(\mathcal C(f)\) stands for the set of points of continuity of a function \(f\). Define \([f=a]=\{x\in\mathbb{R}:f(x)=a\}\); the symbols \([f\neq a]\), \([f>a]\), etc. are defined analogously. A function \(f:\mathbb{R}\to\mathbb{R}\) is called a Baire one star function, if for each nonempty closed set \(P\subset\mathbb{R}\), there is a nonempty portion \(Q=P\cap(a,b)\) of \(P\) such that \(f|_Q\) is continuous. For a given interval \(I\subset\mathbb{R}\) and a function \(\psi:I\to\mathbb{R}\) define by induction on \(\alpha<\omega_1\) NEWLINE\[NEWLINE \mathcal U_\alpha(\psi)=\text{int}\Big(\bigcup_{\beta<\alpha}\mathcal U_\alpha(\psi)\cup\mathcal C\big(\psi|_{I\setminus\bigcup_{\beta<\alpha}\mathcal U_\alpha(\psi)}\big)\Big). NEWLINE\]NEWLINE Let \(\mathcal S_\alpha=\{f:\mathbb{R}\to\mathbb{R}:\mathcal U_\alpha(f)=\mathbb{R}\}\). In particular, \(\mathcal S_0\) is the class of all continuous functions and \(\mathcal S_1\) is the class \(\mathcal B_1^{**}\).NEWLINENEWLINE\textit{A. Lukasiewicz} proved [``A classification of Baire one star functions in topological spaces'', Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 54, No. 1--2, 175--181 (2006; Zbl 1387.54011)] that \(\{\mathcal S_\alpha:\alpha<\omega_1\}\) is a classification of Baire one star functions.NEWLINENEWLINEThe authors' goal is to make the first step toward the characterization of the products of functions from these classes. The main result of the paper is the following.NEWLINENEWLINETheorem. Let \(f:\mathbb{R}\to\mathbb{R}\). Denote by \(\mathcal I\) the family of all bounded connected components \(I=(a,b)\) of \(\mathcal U_0(f)\) with the property that \(f(a)f(b)<0\) and \(I\cap[f=0]=\emptyset\). The following are equivalent:NEWLINENEWLINE(i) there exists \(g,h\in\mathcal S_1\) such that \(f=gh\) on \(\mathbb{R}\);NEWLINENEWLINE(ii) \(f\in\mathcal S_2\) and for each \(x\in[f\neq 0]\setminus\mathcal U_1(f)\): NEWLINE\[NEWLINE \exists\delta>0\;\;\;(x-\delta,x+\delta)\cap\big(\bigcup_{I\in\mathcal I}\text{bd}I\big)'\cap\mathcal U_1(f)=\emptyset, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \forall\varepsilon>0~\exists\delta>0~\forall I\in\mathcal I\big(I\subset[f\cdot f(x)>0]\cap(x-\delta,x+\delta)\Rightarrow\text{dist}(f(x),f[\text{cl}I])<\varepsilon \big). NEWLINE\]NEWLINENEWLINENEWLINEThe proof of this theorem is quite long and it is divided into several auxiliary lemmas. It would be interesting to find a simpler proof of the presented characterization. The authors stated the following open problems.NEWLINENEWLINE1. Given an integer \(n>2\), characterize products of \(n\) functions from \(\mathcal S_1\).NEWLINENEWLINE2. Given nonzero ordinals \(\alpha,\beta<\omega_1\), characterize products of functions \(g\in\mathcal S_\alpha\) and \(h\in\mathcal S_\beta\).NEWLINENEWLINE3. Characterize products of real Baire one double star functions defined on some topological spaces different from \(\mathbb{R}\).
scientific article; zbMATH DE number 6032451