The smallest and the largest families of some classes of \(\mathcal{A}\)-continuous functions (Q6116722)

In the paper properties of \(\mathcal{A}\)-continuous functions (i.e. functions \(f\colon\mathbb{R}\to\mathbb{R}\) such that for every \(x\in\mathbb{R}\) and every neighborhood \(V\) of \(f(x)\) there exists a set \(A\in\mathcal{A}\) containing \(x\) such \(f (A)\subset V\)) are investigated for \(\mathcal{A}\subset 2^\mathbb{R}\). Amongst others, the authors prove that if the family \(\mathcal{A}\) has the \((\mathcal{J}^*)\)-property, i.e. \begin{itemize} \item[(i)] \(\tau_e\subset \mathcal{A}\subset \{G\Delta I\subset \mathbb{R}\colon G\in\tau_e, I\in\mathcal{J}\},\) \item[(ii)] \(A\subset \{x\in\mathbb{R}\colon A\cap G\not\in\mathcal{J}\,\mbox{for every neighbourhood}\; G\; \mbox{of}\; x\}\mbox{ for every}\;\;A\in\mathcal{A},\) \end{itemize} then the family \(\mathcal{C}_\mathcal{A}\) of all \(\mathcal{A}\)-continuous functions satisfies \(\mathcal{C}_{\tau_e}\subset \mathcal{C}_\mathcal{A}\subset C_\mathcal{S}\), where \(\mathcal{S}\) is the family of all semi-open sets. Moreover, families \(\mathcal{A}\) being generalized topologies are considered. More precisely, the formulae describing the smallest and the largest element in the equivalence class \([\mathcal{A}]_\equiv:=\{\mathcal{B}\colon C_\mathcal{A}=C_\mathcal{B}\}\) generated by the generalized topology \(\mathcal{A}\) satisfying special conditions, are given.