Siberian Mathematical Journal

, Volume 47, Issue 2, pp 391–396 | Cite as

On essential ϕ-variation

  • J. Ewert
  • S. P. Ponomarev


We study relations between the classical essential variation and approximate continuity. In particular, we show that the classical essential variation of functions on the so-called τD-regular sets agrees with essential variation in the sense of W. P. Ziemer.


ϕ-variation essential ϕ-variation metric space approximate continuity 


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  1. 1.
    Chistyakov V. V. and Galkin O. E., “Mappings of bounded Φ-variation with an arbitrary function Φ,” J. Dynam. Control Systems, 4, No. 2, 217–247 (1998).MathSciNetGoogle Scholar
  2. 2.
    Cybertowicz Z. and Matuszewska W., “Functions of bounded generalized variation,” Ann. Soc. Math. Polon. Ser. I Comment. Math. Prace Mat., 20, 29–52 (1977).MathSciNetGoogle Scholar
  3. 3.
    Orlicz W., “On functions of finite variation depending on a parameter,” Studia Math., 13, 218–232 (1953).zbMATHMathSciNetGoogle Scholar
  4. 4.
    Ewert J., “Generalized essential variation and quasicontinuity,” Acta Math. Hungar., 108, No. 1–2, 155–160 (2005).zbMATHMathSciNetGoogle Scholar
  5. 5.
    Ponomarev S. P., “Variation preserving extensions and generalized essential variation,” Siberian Math. J., 45, No. 6, 1091–1097 (2004).CrossRefMathSciNetGoogle Scholar
  6. 6.
    Lukeš J., Malý J., and Zajíček L., Fine Topology Methods in Real Analysis and Potential Theory, Springer-Verlag, New York; Berlin; Heidelberg (1986) (Lecture Notes in Math.; V. 1189).Google Scholar
  7. 7.
    Ziemer W. P., Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Springer-Verlag, New York (1989) (Graduate Texts in Math.; V. 120).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • J. Ewert
  • S. P. Ponomarev
    • 1
  1. 1.Institute of MathematicsPedagogical UniversitySlupskPoland

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