Classical Levine's theorem [N. Levine: Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41] asserts that for a semi-continuous mapping on a second countable topological space, the discontinuity points form a 1st category set. There are two directions in literature in which this result is generalized: by considering either multi-valued mappings or mappings on some second noncountable spaces (for the latter, see for instance [T. Neubrunn: Quasi-continuity(topical survey), Real Anal. Exchange 14 (1988/89), 259-306]). In this paper, we offer another path, namely, the path of so-called M-spaces, essentially weaker than the topological ones. Pointwise M -continuity of a mapping between two M-spaces is defined and characterized. These characterizations are the basic tool for our generalization.
