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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The oscillatory behavior of certain derivatives
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by Richard J. O’Malley and Clifford E. Weil PDF
Trans. Amer. Math. Soc. 234 (1977), 467-481 Request permission

Abstract:

The derivatives considered are the approximate derivative and the kth Peano derivative. The main results strengthen the Darboux property, which both of these derivatives possess. Theorem. If the approximate derivative ${f’_{{\text {ap}}}}$ of f exists on an interval and if, for $M \geqslant 0,{f’_{{\text {ap}}}}$ attains both M and — M, then there is an open subinterval where ${f’_{{\text {ap}}}} = f’$ and on which $f’$ attains both M and — M. The other main theorem is obtained from this one by replacing the approximate derivative by the kth Peano derivative.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 234 (1977), 467-481
  • MSC: Primary 26A24
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0453940-3
  • MathSciNet review: 0453940