The set where an approximate derivative is a derivative
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- by Richard J. O’Malley PDF
- Proc. Amer. Math. Soc. 54 (1976), 122-124 Request permission
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
- Casper Goffman and C. J. Neugebauer, On approximate derivatives, Proc. Amer. Math. Soc. 11 (1960), 962–966. MR 118792, DOI 10.1090/S0002-9939-1960-0118792-2
- Richard J. O’Malley, A density property and applications, Trans. Amer. Math. Soc. 199 (1974), 75–87. MR 360955, DOI 10.1090/S0002-9947-1974-0360955-X G. Tolstoff, Sur la dérivée approximative exacte, Mat. Sb. 4 (1938), 499-504.
- Clifford E. Weil, On approximate and Peano derivatives, Proc. Amer. Math. Soc. 20 (1969), 487–490. MR 233944, DOI 10.1090/S0002-9939-1969-0233944-7
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 122-124
- DOI: https://doi.org/10.1090/S0002-9939-1976-0390143-X
- MathSciNet review: 0390143