Abstract
We investigate the problem of construction of a separately continuous function f whose oscillation is equal to a given nonnegative function g. We show that, in the case of a metrizable Baire product, the problem under consideration is solvable if and only if g is upper semicontinuous and its support can be covered by countably many sets, which are locally contained in products of sets of the first category.
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References
R. Kershner, “The continuity of functions of many variables,” Trans. Amer. Math. Soc., 53, No. 1, 83–100 (1943).
Z. Grande, “Une caracterisation des ensembles des points de discontinuite des functions lineairement-continues,” Proc. Amer. Math. Soc., 52, 257–262 (1975).
J. C. Breckenridge and T. Nishiura, “Partial continuity, quasicontinuity, and Baire spaces,” Bull. Inst. Math. Acad. Sinica, 4, No. 2, 191–203 (1976).
V. K. Maslyuchenko and V. V. Mikhailyuk, Separately Continuous Mappings with a Separable Set of Discontinuity Points [in Ukrainian], Dep. in Ukr.NDINTI, No. 907-Uk90, Chernovtsy (1990).
V. K. Maslyuchenko, V. V. Mikhailyuk, and O. V. Sobchuk, “Inverse problems in the theory of separately continuous mappings,” Ukr. Mat. Zh., 44, No. 9, 1209–1220 (1992).
V. K. Maslyuchenko and V. V. Mikhailyuk, “On separately continuous mappings on products of metrizable spaces,” Dop. Akad. Nauk Ukr., No. 28–31 (1993).
V. V. Mikhailyuk, “An inverse problem in the theory of separately continuous mappings. A general approach,” in: Abstracts of the International Mathematical Conference Dedicated to the Memory of Hans Hahn (October, 1994) [in Ukrainian], Ruta, Chernovtsy (1994), p. 104.
V. V. Mikhailyuk, “Characterization of sets of discontinuity points of separately continuous functions on products of metrizable spaces,” in: Abstracts of the International Mathematical Conference Dedicated to the Memory of Hans Hahn (October, 1994) [in Ukrainian], Ruta, Chernovtsy (1994), p. 103.
V. V. Mikhailyuk, Inverse Problems in the Theory of Separately Continuous Mappings [in Ukrainian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Chernovtsy (1994).
R. Engelking, General Topology [Russian translation], Mir, Moscow (1986).
K. Kuratowski, Topology [Russian translation], Vol. 2, Mir, Moscow (1969).
V. K. Maslyuchenko and O. V. Maslyuchenko, “Construction of a separately continuous function with given oscillation,” in: Abstracts of the Scientific Conference Dedicated to the 120 th Anniversary of the Foundation of the Chernovtsy University (May, 1996) [in Ukrainian], Vol. 2, Ruta, Chernovtsy (1995), p. 93.
V. K. Maslyuchenko, “On the characterization of oscillations of separately continuous functions,” in: Abstracts of the All-Ukrainian Conference “Development and Application of Mathematical Methods in Scientific and Technical Investigations” Dedicated to the 70th Birthday of Prof. P. S. Kazimirskii (October, 1995) [in Ukrainian], Part 1, Lvov (1995), pp. 80–81.
V. K. Maslyuchenko, V. V. Mikhailyuk, and O. V. Sobchuk, “Investigations of separately continuous mappings,” in: Abstracts of the International Mathematical Conference Dedicated to the Memory of Hans Hahn [in Ukrainian], Ruta, Chernovtsy (1995), pp. 192–246.
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 948–959, July, 1998.
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Maslyuchenko, V.K., Maslyuchenko, O.V. Construction of a separately continuous function with given oscillation. Ukr Math J 50, 1080–1090 (1998). https://doi.org/10.1007/BF02528836
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DOI: https://doi.org/10.1007/BF02528836