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Transactions of the American Mathematical Society

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



The oscillatory behavior of certain derivatives

Authors: Richard J. O’Malley and Clifford E. Weil
Journal: Trans. Amer. Math. Soc. 234 (1977), 467-481
MSC: Primary 26A24
MathSciNet review: 0453940
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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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Article copyright: © Copyright 1977 American Mathematical Society