Skip to main content
SpringerLink
Log in

Separately continuous functions with respect to a variable frame

  • Brief Communications
  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We show that the set D(f) of discontinuity points of a function f : R 2 → R continuous at every point p with respect to two variable linearly independent directions e 1(p) and e 2(p) is a set of the first category. Furthermore, if f is differentiable along one of directions, then D(f) is a nowhere dense set.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

REFERENCES

  1. R. Baire, “Sur les fonctions de variables réelles,” Ann. Math. Pura Appl., Ser. 3, 3, 1–123 (1899).

    Google Scholar 

  2. R. Kershner, “The continuity of functions of many variables,” Trans. Amer. Math. Soc., 53, No.1, 83–100 (1943).

    Google Scholar 

  3. V. H. Herasymchuk, V. K. Maslyuchenko, and V. V. Mykhailyuk, “Lipschitz varieties and sets of discontinuity points of separately differentiable functions,” Zb. Nauk. Prats’, Ser. Mat., Nauk. Visn. Cherniv. Univ., Issue 134, 22–29 (2002).

  4. Yu. Yu. Trokhymchuk, “On derivatives of functions of many variables along direction,” Ukr. Mat. Zh., 17, No.6, 67–79 (1965).

    Google Scholar 

  5. D. Menchoff, “Sur les différentielles totales des fonctions univalentes,” Math. Ann., 105, 75–85 (1931).

    Article  Google Scholar 

  6. F. W. Gehring and O. Lehto, “On total differentiability of functions of complex variables,” Ann. Acad. Sci. Fenn., Ser. A1, 2–9 (1959).

  7. A. V. Bondar, “On differentiability of open mappings,” Ukr. Mat. Zh., 50, No.12, 1587–1600 (1998).

    Google Scholar 

  8. V. H. Herasymchuk, V. K. Maslyuchenko, and O. V. Maslyuchenko, “Separately continuous and differentiable functions with respect to a variable frame,” in: Abstracts of the International Conference “Complex Analysis and Its Application” (Lviv, May 26–29, 2003) [in Ukrainian], Lviv (2003), pp. 23–24.

  9. V. K. Maslyuchenko, V. V. Mykhailyuk, and V. V. Nesterenko, “Point discontinuity of functions of many variables,” Zb. Nauk. Prats’, Ser. Mat., Nauk. Visn. Cherniv. Univ., Issue 111, 70–76 (2001).

  10. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1968).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ternopil Pedagogic University, Ternopil

    V. H. Herasymchuk

  2. Chernivtsi National University, Chernivtsi

    V. K. Maslyuchenko & O. V. Maslyuchenko

Authors
  1. V. H. Herasymchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. K. Maslyuchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. V. Maslyuchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1281–1286, September, 2004.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Herasymchuk, V.H., Maslyuchenko, V.K. & Maslyuchenko, O.V. Separately continuous functions with respect to a variable frame. Ukr Math J 56, 1526–1531 (2004). https://doi.org/10.1007/s11253-005-0131-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-005-0131-9

Keywords

Search

Navigation