Strong derivatives and integrals (Q2356314)

In this very interesting paper, the author mainly explores the complicated and confusing story of the relations between the Riemann integral and the strong derivative. It is a paper in which the introduction and the interspersed comments are nearly as interesting as the results themselves. It is essential reading for all who think of Saks as the last word on the classes ACG etc., or those who think that Henstock's work is important but impenetrable. The strong derivative at \(x_0\) is obtained by considering the usual ratio using intervals near to but not necessarily straddling \(x_0\). A function \(F\) is strongly differentiable at \(x_0\) if and only if in some neighbourhood of \(x_0\) we have that \(F\) is Lipschitz and that \(F'\) is continuous at \(x_0\), relative to the set of points where it exists. A function is a Riemann primitive if and only if it is Lipschitz and strongly differentiable almost everywhere. By contrast there exist Lebesgue primitives that are nowhere strongly differentiable. One of the difficulties with strong differentiations that there is no Vitali covering theorem related to it. As a result the strong derivative, McShane fine-covers and inner variations do not play a role in the study of Lebesgue's integral that the ordinary derivative and fine covers do in the study of the Henstock-Kurzweil integral. The author does however give proof, using these concepts, of the equivalence of the McShane and Lebesgue integrals.