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Proceedings of the American Mathematical Society
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
MathSciNet review: 0390143
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0390143-X
PII: S 0002-9939(1976)0390143-X
Keywords: Approximate derivative, Baire class 1, Darboux, density
Article copyright: © Copyright 1976 American Mathematical Society