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.


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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