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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Path derivatives: a unified view of certain generalized derivatives

Authors: A. M. Bruckner, R. J. O’Malley and B. S. Thomson
Journal: Trans. Amer. Math. Soc. 283 (1984), 97-125
MSC: Primary 26A24
MathSciNet review: 735410
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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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Keywords: Derivatives, approximate derivatives, Peano derivatives, functions of Baire class $ 1$, Darboux functions
Article copyright: © Copyright 1984 American Mathematical Society