Real Analysis Exchange

The problem of characterizing derivatives revisited

Andy M. Bruckner

Full-text: Open access

Abstract

An outstanding problem is how to characterize the class of derivatives. Our purpose is to discuss the kinds of characterizations one might seek, to indicate the contributions of some attempts to characterize, to look at characterizations that are known, and to discuss the place of the class of derivatives among certain related classes of functions.

Article information

Source
Real Anal. Exchange, Volume 21, Number 1 (1995), 112-133.

Dates
First available in Project Euclid: 3 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1341343228

Mathematical Reviews number (MathSciNet)
MR1377522

Zentralblatt MATH identifier
0998.26500

Subjects
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]
Secondary: 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]

Keywords
derivative Baire 1 approximate continuity

Citation

Bruckner, Andy M. The problem of characterizing derivatives revisited. Real Anal. Exchange 21 (1995), no. 1, 112--133. https://projecteuclid.org/euclid.rae/1341343228


Export citation

References

  • S. A. Agronsky, A generalization of a theorem of Maximoff and applications, Trans. Amer. Math. Soc., 273 no. 2 (1982), 767–779.
  • A. M. Bruckner, Differentiations of Real Functions, CRM Monograph Series, vol. 5, Amer. Math. Soc., Providence, 1994.
  • A. Bruckner J. Ceder and R. Keston, Representations and approximations by Darboux functions in the first class of Baire, Rev. Roum. Math. Pures. Appl., 14 (1968), 1247–1254.
  • A. Bruckner R. O'Malley and B. Thomson, Path Derivatives: A unified view of certain generalized derivatives, Trans. Amer. Math. Soc., 283 (1984), 97–125.
  • A. M. Bruckner and G. Petruska, Some typical results on Bounded Baire $1$ functions, Acta Math. Hung., 43 (1984), 325–333.
  • A. M. Bruckner and B. S. Thomson, Youngs' contributions to real variable theory, Introduction to the collected works of G. C. and W. H. Young, to appear.
  • J. G. Ceder and G. Petruska, Most Darbous Baire $1$ functions map big sets onto small sets, Acta Math. Hung., 41 (1983), 37–46.
  • A. Denjoy, Memoire sur les nombres dérivés des fonctions continues, J. Math. Pures et Appl., 1 no. 7 (1915), 105–240.
  • R. Fleissner, Multiplication and the Fundamental Theorem of Calculus: a survey Real Anal. Ex., 2 (1976), 7–34.
  • C. Freiling, On the problem of characterizing derivatives, to appear.
  • P. Humke and G. Petruska, The packing dimension of a typical function is two, Real Anal. Ex., 14 no. 2 (1988-1989), 345–358.
  • B. Kirchheim, Some further typical results on bounded Baire one functions, Acta Math. Hung., 62 no. 1-2 (1993), 39–49.
  • B. Kirchheim, Typical approximately continuous functions are surprisingly thick, Real Anal. Ex., 18 no. 1 (1992-1993), 55–62.
  • M. Laczkovich, Separation properties of some subclasses of Baire $1$ functions, Acta Math. Acad. Sci. Hung., 26 (1975), 405–412.
  • M. Laczkovich, Separation of sets by derivatives, Acta Math. Acad. Sci. Hung., 27 (1976), 201–207.
  • M. Laczkovich, Separation of sets by bounded derivatives, Period. Math. Hung., 7 (1976), 169–177.
  • M. Laczkovich, A note on approximately continuous and a.e. continuous functions, Acta Math. Acad. Sci. Hung., 33 no. 3-4 (1979), 403–405.
  • J. Lipiński, Sur les fonctions approximativement continues, Colloq. Math., 5 (1958), 172–175.
  • A. Maliszewski, Characteristic functions and products of bounded derivatives, Proc. Amer. Math. Soc., to appear.
  • J. Mařík, Sums of powers of derivatives, Proc. Amer. Math. Soc., 112 no. 3 (1991), 807–817.
  • R. Mauldin, and S. Williams, On the Hausdorff dimension of some graphs, Trans. Amer. Math. Soc., 298 (1986), 793–803.
  • R. Menkyna, Classifying the set where a Baire one function is approximately continuous, Real Anal. Ex., 14 no. 2 (1988-1989), 413–419.
  • M. E. Munroe Introduction to Measure and Integration, Addison-Wesley, Reading, 1953.
  • C. Neugebauer, Darboux functions of Baire class $1$ and derivatives, Proc. Amer. Math. Soc., 13 (1962), 838–843.
  • R. O'Malley and C. Weil, The oscillatory behavior of certain derivatives, Trans. Amer. Math. Soc., 234 (1977), 467–481.
  • G. Petruska, Separation and approximation theorems on derivatives, Acta Math. Acad. Sci. Hung., 25 (1974), 435–442.
  • G. Petruska, On the structure of derivatives, Studia Sci. Math. Hung., 9 (1974), 251–266.
  • G. Petruska, Derivatives take every value in the set of approximate continuity points, Acta Math. Hung., 42 no. 3-4 (1983), 355–360.
  • G. Petruska, and M. Laczkovich, Baire $1$ functions, approximately continuous functions and derivatives, Acta Math. Acad. Sci. Hung., 25 (1974), 189–212.
  • D. Preiss, Maximoff's theorem, Real Anal. Ex., 5 no. 1 (1979-80), 92–104.
  • D. Priess, Algebra generated by derivatives, Real Anal. Ex., 8 no. 1 (1982-83), 205–214.
  • D. Preiss, Level sets of derivatives, Trans. Amer. Math. Soc., 272 no. 1 (1982), 161–184.
  • D. Preiss, and M. Tartaglia, On characterizing derivatives, Proc. Amer. Math. Soc., to appear.
  • A. C. M. von Rooij, On derivatives of functions defined on disconnected sets II, Fund. Math., 131 (1988) 93–102.
  • A. C. M. von Rooij and W. H. Schikhof, On derivatives of functions on disconnected sets I, Fund. Math., 131 (1988), 83–92.
  • W. H. Young, A note on the property of being a differential coefficient, Proc. Lond. Math. Soc., 9 no. 2 (1911), 360–368.
  • W. H. Young, On the differentiation of functions defined by integrals, Trans. Camb. Phil. Soc., 21 (1911), 397–425.
  • Z. Zahorski, Sur la primière dérivée, Trans. Amer. Math. Soc., 69 (1950), 1–54.