On \(\Im\)-a.e. continuous Darboux functions mapping \(\mathbb R^k\) into \(\mathbb R^k\) (Q851073)

The authors investigate relations between classes of functions which are \(\mathfrak I\)-a.e. (almost everywhere) continuous with respect to various \(\sigma\)-ideals~\(\mathfrak I\) of subsets of~\(\mathbb R^k\). They extend the results of \textit{A.~Tomaszewska} [Real Anal. Exch.~25, No. 2, 695--702 (2000; Zbl 1009.26012)] for all powers \(\mathbb R^k\) and also for a~stronger notion of porosity in the metric of uniform convergence. At first they introduce a~notion of a~box-Cantor set in~\(\mathbb R^k\) as a~higher dimensional generalization of symmetric Cantor sets and prove some results for them as well as for the products of ideals. Let \(\mathcal K_k\) denote the \(\sigma\)-ideal of meager subsets of~\(\mathbb R^k\). The following is the main result of the paper: Let \(\mathfrak I\) be a~\(\sigma\)-ideal of subsets of~\(\mathbb R^k\) such that \(\mathfrak I\)~is invariant under homothetic transformations (i.e., \(sA+z\in\mathfrak I\) whenever \(A\in\mathfrak I\), \(z\in R^k\), and \(s\in\mathbb R\)), \(\mathfrak I\)~does not contain any open set, and there exists a~box-Cantor set not belonging to~\(\mathfrak I\). Then the set of all \(\mathfrak I\)-a.e.\ continuous functions is strongly porous in the space of Darboux \(\mathcal K_k\)-a.e.\ continuous functions.