Abstract
Using a game‐theoretic characterization of Baire spaces, conditions upon the domain and the range are given to ensure a fat set C(f) of points of continuity in the sets of type X × {y}, y ∈ Y for certain almost separately continuous functions f : X × Y → Z. These results (especially Theorem B) generalize Mibu′s. First Theorem, previous theorems of the author, answers one of his problems as well as they are closely related to some other results of Debs [1] and Mibu [2].
Highlights
Since the appearance of the celebrated result of Namioka, many articles have been written on the topic of separate and joint continuity, see Piotrowski [3], for a survey Aside from an intensively studied Uniformization Problem- Namioka-type theorems, see Piotrowski [3], questions pertaining to Existence Problem as well as its generalizations, have been asked
One way to ensure the existence of "many" points of continuity in X x Y can be derived from the following
As an immediate consequence we obtain (Piotrowski [4] ..Corollary 4 3) Let X and Y be first countable, Baire spaces and Z be a regular one If f X Y Y is separately continuous, f is symmetrically quasi-continuous If X and Y are second countable Baire spaces and Z is a regular one, and a function f:X x Y Z, the following implications hold see Diagram None of these implications can, in general be replaced by an equivalence, see Neubrunn [5]
Summary
Since the appearance of the celebrated result of Namioka, many articles have been written on the topic of separate and joint continuity, see Piotrowski [3], for a survey Aside from an intensively studied Uniformization Problem- Namioka-type theorems, see Piotrowski [3], questions pertaining to Existence Problem (see below) as well as its generalizations, have been asked. Let us define f 19- IR by f(z, y) 1, for (z, y) E D and f(x, y) 0 if (x, y) D .all the zsections f and all the y-sections f are of first class of Baire and C(f) 05 El. MIBU’S FIRST THEOREM (Mibu [2]) Let X be first countable, Y be Baire and such that. X x Y is Baire Given a metric space M If fXxYM is separately continuous, C(F) is a dense G subset ofX x Y. Let X be second countable, Y be Baire and such that X x Y is Baire Given a metric space M If fXxYM has a) all x-sections f have their sets D(f) of points of discontinuity of the first category and, b) all y-sections f are continuous. X x Y be Baire Given a metric space M If f" X x Y M has: a) all z-sections f of first class- in the sense of Debs and, b) all y-sections fu continuous
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