Real Analysis Exchange

BVp-Functions and Change of Variable

N. Merentes and J. L. Sánchez

Full-text: Open access


In this note we discuss some interconnections between the space \(BV_p[a,b]\) (\(1\leq p\lt\infty\)) of functions of bounded \(p\)-variation (in Wiener's sense) and the space \(Lip_\alpha[a,b]\) (\(0\lt\alpha\leq 1\)) of Hölder continuous functions. In particular, we show that \(f\in BV_p[a,b]\) if and only if \(f=g\circ \tau\), with \(g\in Lip_{1/p}[a,b]\) and \(\tau\) being monotone, and that \(f\in BV_p[a,b] \cap C[a,b]\) if and only if \(f=g\circ \tau\), with \(g\in Lip_{1/p}[a,b]\) and \(\tau\) being a homeomorphism.

Article information

Real Anal. Exchange, Volume 37, Number 1 (2011), 177-188.

First available in Project Euclid: 30 April 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Primary: 26A45: Functions of bounded variation, generalizations 26A16: Lipschitz (Hölder) classes
Secondary: 26A48: Monotonic functions, generalizations

bounded \(p\)-variation Hölder continuity homeomorphism, change of variables


Merentes, N.; Sánchez, J. L. BV p -Functions and Change of Variable. Real Anal. Exchange 37 (2011), no. 1, 177--188.

Export citation


  • J. Appell, E. D'Aniello, M. Väth, Some remarks on small sets, Ric. Mat. 50 (2001), 255–274.
  • A. N. Bakhvalov, M. I. Dyachenko, K. S. Kazaryan, P. Sifuentes, P. L. Ul'yanov, Real Analysis in Exercises, Fizmatlit, Moskva 2005. (Russian).
  • A. M. Bruckner, C. Goffman, Differentiability through change of variables, Proc. Amer. Math. Soc., 61 (1976), 235–241.
  • M. Bruneau, Variation Totale d'une Fonction, Lecture Notes in Math. 413, Springer, Berlin 1974.
  • F. W. Gehring, A study of $\alpha$-variation, Trans. Amer. Math. Soc., 76 (1954), 420–443.
  • B. R. Gelbaum, J. M. H. Olmstedt, Counterexamples in Analysis, Holden-Day, San Francisco, 1964.
  • F. N. Huggins, Some interesting properties of the variation function, Amer. Math. Monthly, 83 (1976), 538–546.
  • C. Jordan, Sur les séries de Fourier, C. R. Math. Acad. Sci. Paris 2 (1881), 228–230.
  • R. Kannan, C. K. Krueger, Advanced Analysis on the Real Line, Springer, Berlin 1996.
  • E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1934), 837–842.
  • Z. Zahorski, Über die Menge der Punkte, in welchen die Ableitung unendlich ist, Tohoku Math. J. 48 (1941), 321–330.
  • Z. Zahorski, Sur la première dérivée, Trans. Amer. Math. Soc., 69 (1950), 1–54.