On Properties of Derivatives

Introduction. Although derivatives need not be continuous, the inverse images of open intervals are heavy in another sense. Denjoy and Clarkson have shown that the inverse image of every open interval either is empty or has positive measure (see [l] and [3]). Zahorski refined this property to one of homogeneity. He proved that if x is in the inverse image of an open interval (a, b), and if { /„ j is a sequence of closed intervals converging to x (every neighborhood of x contains all but a finite number of the closed intervals /„) such that each of the closed intervals essentially misses the inverse image of (a, b), then the ratio of the length of /„ to the distance of /„from x converges to 0 as n increases (see [9]). These properties together with Baire one and the Darboux property (takes connected sets to connected sets) do not classify derivatives. In fact, it is established here that these two properties are possessed by every approximate derivative and by every fcth Peano derivative.