Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Limits of differentiable functions


Author: Udayan B. Darji
Journal: Proc. Amer. Math. Soc. 124 (1996), 129-134
MSC (1991): Primary 26A24, 26A21; Secondary 40A30
DOI: https://doi.org/10.1090/S0002-9939-96-02998-X
MathSciNet review: 1285985
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $\{f_n\}$ is a sequence of differentiable functions defined on [0,1] which converges uniformly to some differentiable function $f$, and $\{f'_n\}$ converges pointwise to some function $g$. Let $M = \{x: f'(x) \neq g(x)\}$. In this paper we characterize such sets $M$ under various hypotheses. It follows from one of our characterizations that $M$ can be the entire interval [0,1].


References [Enhancements On Off] (What's this?)

  • 1 Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. MR 507448
  • 2 Krzysztof Ciesielski, Lee Larson, and Krzysztof Ostaszewski, ℐ-density continuous functions, Mem. Amer. Math. Soc. 107 (1994), no. 515, xiv+133. MR 1188595, https://doi.org/10.1090/memo/0515
  • 3 R. Jeffery, The theory of function of a real variable, Mathematical Exposition No. 6, University of Toronto Press, Ontario, 1951. MR 13:2166
  • 4 K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR 0217751
  • 5 David Preiss, Limits of approximately continuous functions, Czechoslovak Math. J. 21 (96) (1971), 371–372. MR 0286947
  • 6 Walter Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. International Series in Pure and Applied Mathematics. MR 0385023
  • 7 Z. Zahorski, Sur la premiere derivee, Trans. Amer. Math. Soc. 69 (1950), 1--54.MR 12:247c

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 26A24, 26A21, 40A30

Retrieve articles in all journals with MSC (1991): 26A24, 26A21, 40A30


Additional Information

Udayan B. Darji
Affiliation: Department of Mathematics University of Louisville Louisville, Kentucky 40292
Email: ubdarj01@homer.louisville.edu

DOI: https://doi.org/10.1090/S0002-9939-96-02998-X
Keywords: $F_\sigma$, $G_{\delta\sigma}$, density topology, approximate continuity, nowhere measure dense
Additional Notes: This is the core part of the author’s dissertation which was directed by Professor Jack B. Brown
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society