Abstract
The present paper investigates the problem of constructing a separately continuous function defined on the product of two topological spaces that possesses a specified set of points of discontinuity and the related special problem of constructing a pointwise convergent sequence of continuous functions that possesses a specified set of points of nonuniform convergence and set of points of discontinuity of a limit function. In the metrizable case the former problem is solved for separable Fσ-sets whose projections onto every cofactor is of the first category. The second problem is solved for a pair of embedded Fσ.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Namioka, “Separate continuity and joint continuity,” Pacif. J. Math.,51, No. 2, 515–531 (1974).
G. Young and W. Young, “Discontinuous functions continuous with respect to every straight line,” Quart. J. Pure Appl. Math.,41, 8793 (1909).
R. Kershner, “The continuity of functions of many variables,” Trans. Amer. Math. Soc.,53, 83–100 (1943).
R. E. Feiock, “Cluster sets and joint continuity,” J. London Math. Soc.,7, 397–406 (1973).
Z. Grande, “A description sets of points of discontinuity of linearly continuous functions,” Proc. Amer. Math. Soc.,52, 257–262 (1975).
M. M. Mirzoyan, “On limit sets of mappings of topological spaces,” Dokl. Akad. Nauk SSSR,243, No. 1, 37–40 (1978).
M. Ya. Tseitlin, “On the set of points of discontinuity of separately continuous functions,” in: Measure and Integral [in Russian], Kuibyshev (1988), 147–151.
H. Hahn, Reelle Funktionen. 1. Teil. Punktfunktionen, Akad. Verlagsgesellschaft, Leipzig (1932), 415 pp.
J. C. Breckenridge and T. Nishiuza, “Partial continuity, quasicontinuity, and Baire spaces,” Bull. Inst. Math. Acad. Sinica,4, No. 2, 191–203 (1976).
Z. Piotrowski, “Separate and joint continuity. II,” Real Anal. Exch.,15, No. 1, 248–258 (1989–1990).
Z. Piotrowski, “Separate and joint continuity,” Real Anal. Exch.,11, No. 2, 293–322 (1985–1986).
P. Engelking, General Topology [Russian translation], Mir, Moscow (1986), 751 pp.
Ya. I. Grushka, V. K. Maslyuchenko, and O. V. Sobchuk, On Baire's and Osgood's Theorems [in Ukrainian], Chernivtsi (1990), 12 pp. (deposited at UkrNDINTI, No. 903-Uk90).
V. K. Maslyuchenko and V. V. Mikhailyuk, Separately Continuous Functions with Separable Set of Points of Discontinuity [in Ukrainian], Chernivtsi (1990), 11 pp. (deposited at UkrNDINTI, No. 90-Uk90).
J. Oxtoby, Measure and Category [Russian translation], Mir, Moscow (1974), 158 pp.
Author information
Authors and Affiliations
Additional information
Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1209–1220, September, 1992.
Rights and permissions
About this article
Cite this article
Maslyuchenko, V.K., Mykhaylyuk, V.V. & Sobchuk, O.V. Inverse problems of the theory of separately continuous mappings. Ukr Math J 44, 1108–1116 (1992). https://doi.org/10.1007/BF01058371
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01058371