Holá, Ľubica; Piotrowski, Zbigniew Set of continuity points of functions with values in generalized metric spaces. (English) Zbl 1212.54040 Tatra Mt. Math. Publ. 42, 149-160 (2009). The paper considers the continuity points of functions with values in generalized metric spaces. For a function targeted to a weakly developable space, the notion of generalized oscillation is defined as a substitute for the oscillation (they can differ in metrizable spaces). A function with values in a weakly developable space is continuous at a point exactly when its generalized oscillation at this point is 0. The set of continuity points of such a function is \(G_\delta \). Another result obtained is that a topological space has a \(G_\delta \)-diagonal whenever it has the above property for any function targeted to it.The set of continuity points of a quasicontinuous function from a Baire space to a weakly developable space is dense \(G_\delta \). This is used to prove that, if \(X\) is first countable, \(Y\) is Baire and \(Z\) is a regular weakly developable space, then, for any \(f\:X\times Y\to Z\) with quasicontinuous sections \(f_x\) and continuous sections \(f_y\), the set of continuity points of \(f\) lying in \(\{x\}\times Y\) is dense \(G_\delta \) therein thus extending a result for \(Z\) a Moore space.The set of continuity points of a function with a closed graph to a \(p\)-space \(Y\) is shown to be \(G_\delta \). This remains true if \(Y\) is a \(w\Delta \)-space, provided that the domain space is first countable. If \(f\) is a continuous bijective map from a hereditarily Baire \(p\)-space, such that the points of continuity of \(f^{-1}\) form a dense set, then the target space contains a dense Baire subspace (so it is Baire itself). Reviewer: Ivaylo S. Kortezov (Sofia) Cited in 7 Documents MSC: 54C05 Continuous maps 54C08 Weak and generalized continuity 54E52 Baire category, Baire spaces Keywords:\(p\)-space; \(G_\delta \)-diagonal; developable space; weakly developable space; quasicontinuous function; generalized oscillation PDFBibTeX XMLCite \textit{Ľ. Holá} and \textit{Z. Piotrowski}, Tatra Mt. Math. Publ. 42, 149--160 (2009; Zbl 1212.54040)