Path derivatives: a unified view of certain generalized derivatives

Abstract

A collection E = { E x : x ∈ R } E = \{ {E_x}:x \in R\} is a system of paths if each set E x {E_x} has x x as a point of accumulation. For such a system E E the derivative F E ′ ( x ) F_E’(x) of a function F F at a point x x is just the usual derivative at x x relative to the set E x {E_x} . The goal of this paper is the investigation of properties that F F and its derivative F E ′ F_E’ must have under certain natural assumptions about the collection E 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 E relating to the "thickness" of the sets E x {E_x} and the way in which the sets intersect.