Classical Levine's theorem [N. Levine: \textit{Semi-open sets and semi-continuity in topological spaces}, Amer. Math. Monthly{\bf70} (1963), 36--41] asserts that for a semi-continuous mapping on a second countable topological space, the discontinuity pointsform a 1st category set. There are two directions in~literature in which this result is generalized: by considering either multi-valuedmappings or mappings on some second noncountable spaces (for the latter, see for instance [T.~Neubrunn:\textit{Quasi-continuity (topical survey)}, Real Anal. Exchange {\bf14} (1988/89), 259--306]). In this paper,we offer yet another path, namely, the path of so-called $\mathpzc{M}$-spaces, essentially weaker than the topological ones.Pointwise $\mathpzc{M}$-continuity of a mapping between two $\mathpzc{M}$-spaces is defined and characterized.These characterizations are the basic tool for our generalization.
@article{176,
title = {On pointwise $\mathpzc{M}$-continuity of mappings},
journal = {Tatra Mountains Mathematical Publications},
volume = {51},
year = {2012},
doi = {10.2478/tatra.v52i0.176},
language = {EN},
url = {http://dml.mathdoc.fr/item/176}
}
Duszyński, Zbigniew. On pointwise $\mathpzc{M}$-continuity of mappings. Tatra Mountains Mathematical Publications, Tome 51 (2012) . doi : 10.2478/tatra.v52i0.176. http://gdmltest.u-ga.fr/item/176/