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)

 
 

 

The set where an approximate derivative is a derivative


Author: Richard J. O’Malley
Journal: Proc. Amer. Math. Soc. 54 (1976), 122-124
DOI: https://doi.org/10.1090/S0002-9939-1976-0390143-X
MathSciNet review: 0390143
Full-text PDF

Abstract | References | Additional Information

Abstract: Let $ f:[0,1] \to R$ possess a finite approximate derivative $ f_{\operatorname{ap}}'$ Let $ E$ be the set of points $ x$ where $ f$ is actually differentiable. It is shown that for every $ \lambda $ if $ \{ x:f_{\operatorname{ap}}'(x) = \lambda \} \ne \emptyset $, then $ \{ x:f_{\operatorname{ap}}'(x) = \lambda \} \cap E \ne \emptyset $. A strengthening of the mean value theorem associated with approximate derivatives is an immediate corollary.


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

  • [1] C. Goffman and C. J. Neugebauer, On approximate derivatives, Proc. Amer. Math. Soc. 11 (1960), 962-966. MR 22 #9562. MR 0118792 (22:9562)
  • [2] R. J. O'Malley, A density property with applications, Trans. Amer. Math. Soc. 199 (1974), 75-87. MR 0360955 (50:13402)
  • [3] G. Tolstoff, Sur la dérivée approximative exacte, Mat. Sb. 4 (1938), 499-504.
  • [4] C. E. Weil, On approximate and Peano derivatives, Proc. Amer. Math. Soc. 20 (1969), 487-490. MR 38 #2265. MR 0233944 (38:2265)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0390143-X
Keywords: Approximate derivative, Baire class 1, Darboux, density
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society