Analogues of transitiveness and decomposition of continuity (Q1704441)

There are many articles about decompositions of continuity. The paper under review is a part of this trend of research. The authors continue their considerations initiated in [Mat. Visn. Nauk. Tov. Im. Shevchenka 8, 132--150 (2011; Zbl 1265.54089)]. In that article, they introduced the notion of transitiveness and used it in a theorem on decomposition of continuity. Recall that a function \(f:X\to Y\) is called \textit{transitional} at \(x\in X\) if, for each neighborhood \(V\) of \(f(x) \in Y\), there is a neighborhood \(U\) of \(x\) and an open neighborhood \(W\) of \(f(x)\) such that \(W\subset V\) and \(U \cap f^{-1}(\mathrm{fr }W) = \emptyset\) (where fr denotes the boundary). Now, in the first part of the paper under review the authors define several properties which are similar to transitiveness and investigate relationships between them. In the second part the authors prove a local version of the theorem on decomposition of continuity from the previous paper and some similar results.