The oscillatory behavior of certain derivatives

O’Malley, Richard J.; Weil, Clifford E.

doi:10.1090/S0002-9947-1977-0453940-3

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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