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Points of joint continuity and large oscillations

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Ukrainian Mathematical Journal Aims and scope

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 B (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 ε(f) of all points p ∈ X × Y at which the oscillation ω f (p) ≥ ε onto X is a closed set nowhere dense in X.

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Authors and Affiliations

  1. Chernivtsi National University, Chernivtsi, Ukraine

    V. K. Maslyuchenko & V. V. Nesterenko

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  1. V. K. Maslyuchenko
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  2. V. V. Nesterenko
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 6, pp. 791–800, June, 2010.

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Maslyuchenko, V.K., Nesterenko, V.V. Points of joint continuity and large oscillations. Ukr Math J 62, 916–927 (2010). https://doi.org/10.1007/s11253-010-0400-0

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  • DOI: https://doi.org/10.1007/s11253-010-0400-0

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