The right absorption property for Darboux functions (Q1898964)

Let \(\mathcal F\) be a family of Darboux surjections mapping \(\mathbb{R}\) onto \(\mathbb{R}\). \(\mathcal F\) has the \textit{Darboux right absorption property} relative to a topological space \(Z\) (abbreviated \(DRAP(Z)\)) provided that, if \(g: \mathbb{R}\to Z\) is such that \(g\circ f\) is a Darboux function for some \(f\in {\mathcal F}\), then \(g\) is also a Darboux function. The author investigates the problem of characterization of the maximal class \(\mathcal F\) with the property \(DRAP(Z)\) (this family is denoted by \({\mathcal F}_D\)). This problem remains unsolved, nevertheless the author shows that if \(Z\) is a continuous image of \(\mathbb{R}\) then \(\widehat C\subset {\mathcal F}_D\subset D^*_{\mathbb{R}}\) and \(\widehat C\neq {\mathcal F}_D\neq D^*_{\mathbb{R}}\), where: a Darboux surjection \(f: \mathbb{R}\to \mathbb{R}\) belongs to the family \(\widehat C\) if for every \(y\in \mathbb{R}\) there exists a non- degenerate compact interval \(I_y\subset \mathbb{R}\) such that \(f|I_y\) is continuous and \(y\in \text{int } f(I_y)\); a surjection \(f: \mathbb{R}\to \mathbb{R}\) belongs to the family \(D^*_{\mathbb{R}}\) if for every \(y\in \mathbb{R}\) and a neighbourhood \(V\) of \(y\) there exists \(x\in f^{- 1}(y)\) and an unilateral neighbourhood \(U\) of \(x\) such that \(f[U]\subset V\).