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Banach Category Theorem; categorical almost continuity; Blumberg space; separate and joint continuity
Based on some earlier findings on Banach Category Theorem for some "nice" $\sigma$-ideals by J. Kaniewski, D. Rose and myself I introduce the $h$ operator ($h$ stands for "heavy points") to refine and generalize kernel constructions of A. H. Stone. Having obtained in this way a generalized Kuratowski's decomposition theorem I prove some characterizations of the domains of functions having "many" points of $h$-continuity. Results of this type lead, in the case of the $\sigma$-ideal of meager sets, to important statements of Abstract Analysis such as Blumberg or Namioka-type theorems.
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