On some properties connected with continuity and measurability (Q2774506)

In this paper the authors introduce some generalizations of the continuity and cliquishness of functions. These generalizations are connected with some (set theoretical) properties \(\mathcal W(x)\) and \(\mathcal U(x)\) which are connected with a fixed family \(\mathcal W\) of subsets of the domain of the function. NEWLINENEWLINENEWLINEIn the further part the authors consider a fixed \(\sigma \)-ideal \(\mathcal J\) of subsets of the domain and the smallest \(\sigma \)-field \(\mathcal M\) containing all sets from \(\mathcal W\cup \mathcal J\). The investigations are connected with the \(\mathcal M\) measurability of functions.