Abstract
It is proved that, for a metric space X and a normed space Z, the diagonals of pointwise Lipschitz mappings f : X 2 → Z are exactly stable pointwise limits of pointwise Lipschitz mappings. The joint Lipschitz property of separately pointwise Lipschitz mappings f : X × Y → Z, where X, Y, and Z are metric spaces, is investigated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Baire, “Sur les fonctions de variables reélles,” Ann. Mat. Pura Appl., 3, No. 3, 1–123 (1899).
T. Banakh, “(Metrically) Quater-stratifable spaces and their applications in the theory of separately continuous functions,” Mat. Stud., 18, No. 1, 10–18 (2002).
K. Borsuk, Theory of Retracts, Panstw. Wydawn. Naukowe, Warsaw, 1967.
M. Burke, “Borel measurability of separately continuous functions,” Top. Appl., 129, No. 1, 29–65 (2003).
J. Dugundji, “An extension of Tietze’s theorem,” Pacific. J. Math., 1, 353–367 (1951).
R. Engelking, General Topology, Panstw. Wydawn. Naukowe, Warsaw, 1977.
G. N. Fikhtengol’ts, Course of Differential and Integral Calculi [in Russian], GITTL, Moscow, 1951, Vol. 2.
V. G. Gerasymchuk, V. K. Maslyuchenko, and V. V. Mykhaylyuk, “Varieties of the Lipschitz property and sets of the discontinuity points of separately differentiable functions,” Nauk. Visn. ChNU. Mat., 134, 22–29 (2002).
H. Hahn, Theorie der Reellen Funktionen, Band 1, Springer, Berlin, 1921.
H. Lebesque, “Sur l’approximation des fonctions,” Bull. Sci. Math., 22, 278–287 (1898).
H. Lebesque, “Sur les fonctions representables analytiquement,” J. de Math., 2, No. 1, 139–216 (1905).
V. K. Maslyuchenko, V. V. Mykhaylyuk, and O. V. Sobchuk, “Construction of a separately continuous function of n variables with given diagonal,” Mat. Studii, 12, No. 1, 101–107 (1999).
O. V. Maslyuchenko, V. K. Maslyuchenko, V. V. Mykhaylyuk, and O. V. Sobchuk, “Paracompactness and separately continuous mappings,” in: General Topology in Banach spaces, Nova Sci., New York, 2001, pp. 147–169.
W. Moran, “Separate continuity and support of measures,” J. London. Math. Soc., 44, 320–324 (1969).
V. V. Mykhaylyuk, “Construction of separately continuous functions of n variables with given restriction,” Ukr. Mat. Visn., 3, No. 3, 374–381 (2006).
W. Rudin, “Lebesgue’s first theorem,” Adv. in Math. Suppl. Stud., 7B, 741–747 (1981).
G. Vera, “Baire measurability of separately continuous functions,” Quart. J. Math. Oxford, 39, No. 153, 109–116 (1988).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 10, No. 3, pp. 343–359, July–August, 2013.
Rights and permissions
About this article
Cite this article
Mykhaylyuk, V.V., Sobchuk, O.V. Diagonals of separately pointwise Lipschitz mappings. J Math Sci 196, 652–664 (2014). https://doi.org/10.1007/s10958-014-1683-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1683-8