Various continuities with the density, \(\mathcal I\)-density and ordinary topologies on \(\mathbb{R}\) (Q1189084)

The aim of the paper is to investigate sixteen classes of continuous real functions of a real variable which are obtained when the domain and the range are equipped with one of the following topologies: the ordinary topology, density topology, \(I\)-density topology and deep \(I\)-density topology. It turns out that some of these classes consist only of constant functions. The inclusions between more interesting classes are presented in several diagrams. Also the relationship between them and analytic, \(C^ \infty\), Baire*1 and Baire 1 functions is studied. The proofs are developed with the techniques prepared in earlier papers by the same authors.