Limits of differentiable functions
HTML articles powered by AMS MathViewer
- by Udayan B. Darji PDF
- Proc. Amer. Math. Soc. 124 (1996), 129-134 Request permission
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
- Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. MR 507448, DOI 10.1007/BFb0069821
- Krzysztof Ciesielski, Lee Larson, and Krzysztof Ostaszewski, $\scr I$-density continuous functions, Mem. Amer. Math. Soc. 107 (1994), no. 515, xiv+133. MR 1188595, DOI 10.1090/memo/0515
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- David Preiss, Limits of approximately continuous functions, Czechoslovak Math. J. 21(96) (1971), 371–372. MR 286947, DOI 10.21136/CMJ.1971.101036
- Walter Rudin, Principles of mathematical analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. MR 0385023
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Additional Information
- Udayan B. Darji
- Affiliation: Department of Mathematics University of Louisville Louisville, Kentucky 40292
- MR Author ID: 318780
- ORCID: 0000-0002-2899-919X
- Email: ubdarj01@homer.louisville.edu
- 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
- © Copyright 1996 American Mathematical Society
- 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