Points of joint continuity and large oscillations

Maslyuchenko, V.; Nesterenko, V.
December 2010
Ukrainian Mathematical Journal;Dec2010, Vol. 62 Issue 6, p916
Academic Journal
For topological spaces X and Y and a metric space Z, we introduce a new class $ \mathcal{N}\left( {X \times Y,\,Z} \right) $ of mappings f: X × Y → Z containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping f from this class and any countable-type set B in Y, the set C( f) of all points x from X such that f is jointly continuous at any point of the set { x} × B is residual in X: We also prove that if X is a Baire space, Y is a metrizable compact set, Z is a metric space, and $ f \in \mathcal{N}\left( {X \times Y,\,Z} \right) $, then, for any ε > 0, the projection of the set D( f) of all points p ∈ X × Y at which the oscillation ω( p) ≥ ε onto X is a closed set nowhere dense in X.


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