The oscillatory behavior of certain derivatives
Richard J. O’Malley, Clifford E. Weil · Transactions of the American Mathematical Society · 1977
The derivatives considered are the approximate derivative and the k th Peano derivative. The main results strengthen the Darboux property, which both of these derivatives possess. Theorem. If the approximate derivative f ap ′ {f’_{{\text {ap}}}} of f exists on an interval and if, for M ⩾ 0 , f ap ′ M \geqslant 0,{f’_{{\text {ap}}}} attains both M and — M, then there is an open subinterval where f ap ′ = f ′ {f’_{{\text {ap}}}} = f’ and on which f ′ f’ attains both M and — M . The other main theorem is obtained from this one by replacing the approximate derivative by the k th Peano derivative.