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Extreme results on certain generalized Riemann derivatives. (English) Zbl 1391.26019

Let \(r\) be a positive integer. The main result of the paper under review establishes that for certain generalized Riemann derivatives of order \(r\), there exists a function \(f:{\mathbb R}\rightarrow {\mathbb R}\) which is \((r-1)\)-times continuously differentiable and such that this \(r\)th generalized derivative does not exist on the whole real axis. A related result of this paper establishes that such a function \(f\) exists for the classical \(r\)th Riemann derivative and if \(r\) is an odd positive integer.

MSC:

26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
11A55 Continued fractions
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A51 Convexity of real functions in one variable, generalizations
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