On some properties of the class \(\mathcal A^\star\) (Q1852450)

For \(x,y\in I=[0,1]\) let \(I_{(x,y)}\) denote the closed interval with the ends \(x\) and \(y\). For a function \(f\colon I\to I\) and for \(x,y\in I\) let \(A^f_{xy}= I_{(f(x),f(y))}\setminus f(I_{(x,y)})\) and \(A^f=\bigcup_{x,y\in I}A^f_{xy}\). Let \(\mathcal J\) be the family of all \(\sigma\)-ideals of boundary subsets of \(I\) which contain all singletons. For \(J\in {\mathcal J}\) let \({\mathcal A}^{\star}_{J}\) denote the family of all nowhere constant functions \(f\colon I\to I\) with \(A^f\in J\). Then let \({\mathcal A}^{\star}=\bigcup_{J\in {\mathcal J}}{\mathcal A}^{\star}_{J}\). Moreover, for \(J\in {\mathcal J}\) let \({\mathcal Q}_{\mathcal D}^{J}\) denote the family of all nowhere constant functions \(f\colon I\to I\) such that \(A^f_{xy}\in J\) for every \(x,y\in I\). Then let \({\mathcal Q}_{\mathcal D}=\bigcup_{J\in{\mathcal J}}{\mathcal Q}_{\mathcal D}^{J}\). It is easy to observe that \({\mathcal A}^{\star}\subset {\mathcal Q}_{\mathcal D}\). The main result of the first part of the paper is the equality \({\mathcal A}^{\star}={\mathcal Q}_{\mathcal D}\). In the second part of the paper the authors work with the space of real-valued functions defined on the space \({\mathcal A}^{\star}\). The results are connected with the class \({\mathcal D}_{\mathcal P}\) of all functions \(f\colon {\mathcal A}^{\star}\to {\mathbb{R}}\) such that \(f(L)\) is a connected set for each arc \(L\subset {\mathcal A}^{\star}\). It is proven that: (1) There exists \(\Phi\subset {\mathcal A}^{\star}\) such that if \(f\in{\mathcal D}_{\mathcal P}\) and \(f|\Phi\) or \(f|({\mathcal A}^{\star}\setminus\Phi)\) is quasi continuous (cliquish), \(f\) is so (but the corresponding assertion for continuity is false). (2) The set \({\mathcal D}_{\mathcal P}\) is porous at each point of some big subset of the space of all functions \(f\colon {\mathcal A}^{\star}\to{\mathbb{R}}\) (with the metric of uniform convergence).