On approximate and Peano derivatives

Abstract

Introduction. It has been known for some time that the inverse image of an open interval under a derivative either is empty or has positive measure (see [1] or [2]). Somewhat later it was shown that this property is also possessed by every approximate derivative and every kth Peano derivative (see [4], [5] or [6]). At that time it was known that an approximate derivative is an ordinary derivative on an open dense set (see [3]), and likewise, a kth Peano derivative is an ordinary kth derivative on an open dense set (see [5]). This paper proves that these ordinary derivatives possess the above property on their sets of existence. DEFINITIONS. Throughout, all functions are real valued functions defined on some fixed connected subset of the real line. Lebesgue measure on the line will be denoted by u. DEFINITION 1. A function f has an approximate derivative fap if for each x in its domain there is a set E whose density (computed relative to the domain of f) at x is 1 such that