On some class of functions intermediate between the class \(B^*_1\) and the family of continuous functions (Q2774507)

We say that \(f\:\mathbb R\to \mathbb R\) belongs to the class \(B^{**}_1\) if either \(D_{f}=\emptyset \) or \(f|D_{f}\) is a continuous function (where \(D_f\) denotes the set of all discontinuity points of \(f\)). NEWLINENEWLINENEWLINEThe topology \(\mathcal T\) is said to be compatible with the natural topology of the real line (\(\mathcal T_0\)) relative to the family of functions \(\mathcal F\) if NEWLINENEWLINENEWLINE1. \(f\:(\mathbb R,\mathcal T)\to \mathbb R\) is a continuous function for any \(f\in \mathcal F\). NEWLINENEWLINENEWLINE2. If \(\mathcal T_1\) and \(\mathcal T_2\) are topologies of the subspace \(\mathbb R\setminus \bigcup _{f\in \mathcal F}\overline {D_f}\) of the space \((\mathbb R,\mathcal T)\) and \(\mathbb R\), respectively; then \(\mathcal T_1 =\mathcal T_2\). NEWLINENEWLINENEWLINE3. For any \(x\in \mathbb R\) and each \(\mathcal T\)-neighbourhood \(U\) of \(x\), \(x\) is a bilateral accumulation point of \(U\setminus \bigcup _{f\in \mathcal F}\overline {D_f}\) (in the topology \(\mathcal T_0\)). NEWLINENEWLINENEWLINEThe class \(\Phi \) is an ordinary system (in the sense of Aumann) if NEWLINENEWLINENEWLINE1. \(f,g\in \Phi \) imply \(\max \{f,g\} \in \Phi \), \(\min \{f,g\} \in \Phi \). NEWLINENEWLINENEWLINE2. \(\Phi \) contains all constants. NEWLINENEWLINENEWLINE3. \(f,g \in \Phi \) imply \(f+g \in \Phi \), \(f\cdot g \in \Phi \) and \(f/g \in \Phi \), for \(g\) such that \(\{x\: g(x)=0\}=\emptyset \). NEWLINENEWLINENEWLINEThe following main result is included in this paper: NEWLINENEWLINENEWLINELet \(f,g \in DB^{**}_1\) (\(DB^{**}_1\) denote the class of Darboux functions belonging to the class \(B^{**}_1\) at the same time) be the functions such that \(D_f=D_g\). Then the following conditions are equivalent: NEWLINENEWLINENEWLINE(a) \(f,g \in \Phi \), where \(\Phi \subset DB^{**}_1\) is some complete system containing each continuous function; NEWLINENEWLINENEWLINE(b) \(f,g \in \Phi \), where \(\Phi \subset DB^{**}_1\) is some ordinary system containing each continuous function; NEWLINENEWLINENEWLINE(c) there exists a topology \(\mathcal T\) finer than the natural topology of the real line \(\mathcal T_0\) and compatible with the topology \(\mathcal T_0\) relative to the family \({f,g}\).