Abstract
We investigate the existence of a separately continuous function f: X × Y → ℝ with a one-point set of discontinuity points in the case where the topological spaces X and Y satisfy conditions of compactness type. In particular, it is shown that, for compact spaces X and Y and nonisolated points x 0 ∈ X and y 0 ∈ Y, a separately continuous function f: X × Y → ℝ with the set of discontinuity points {(x 0, y 0)} exists if and only if there exist sequences of nonempty functionally open sets in X and Y that converge to x 0 and y 0, respectively.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 94–101, January, 2005.
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Mykhailyuk, V.V. One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces. Ukr Math J 57, 112–120 (2005). https://doi.org/10.1007/s11253-005-0174-y
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DOI: https://doi.org/10.1007/s11253-005-0174-y