Path derivatives: a unified view of certain generalized derivatives

Bruckner, A. M.; O’Malley, R. J.; Thomson, B. S.

doi:10.1090/S0002-9947-1984-0735410-1

Path derivatives: a unified view of certain generalized derivatives
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by A. M. Bruckner, R. J. O’Malley and B. S. Thomson PDF

Trans. Amer. Math. Soc. 283 (1984), 97-125 Request permission

Abstract:

A collection $E = \{ {E_x}:x \in R\}$ is a system of paths if each set ${E_x}$ has $x$ as a point of accumulation. For such a system $E$ the derivative $F_E’(x)$ of a function $F$ at a point $x$ is just the usual derivative at $x$ relative to the set ${E_x}$. The goal of this paper is the investigation of properties that $F$ and its derivative $F_E’$ must have under certain natural assumptions about the collection $E$. In particular, it is shown that most of the familiar properties of approximate derivatives and approximately differentiable functions follow in this setting from three conditions on the collection $E$ relating to the "thickness" of the sets ${E_x}$ and the way in which the sets intersect.

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