On the Baire Class of J-Approximate Derivatives

Abstract

Basic properties of an f approximate derivative which is a category analogue of the approximate derivative were considered in [3]. Using the notion of balanced selection and the result of O'Malley, we shall prove that the f approximate derivative is of Baire class one. Throughout the paper, V will denote the family of all subsets of R having the Baire property and f the a-ideal of sets of the first category. For a C R and A c R we denote a * A = {ax: x e A) and A a = {x a: x E A}. Recall [4] that 0 is an f-density point of a set A E B if and only if xn.An[-,1 1 as n -* , i.e. if and only if for every increasing sequence { nm } m N Of natural numbers there exists a subsequence { nnt }p E N such that Xn. A n-1,1] -1 asp -oo except for a set belonging to J. A point x0 is an f-density point of A E E if and only if 0 is an f-density point of A xo. A point x0 is an f-dispersion point of A E E' if and only if x0 is an f-density point of R \A. The notions of right-hand and left-hand f-density points are defined in an obvious manner. DEFINITION. Let F be any finite function defined in some neighborhood of x0, and having there the Baire property, and let C(x, X = F(x) F(x0) for x A x0. We define the J-approximate upper derivative (F,>ap(xo)) as the greatest lower bound of the set { a: {x: C(x, xo) > a) has x0 as an s-dispersion point}. The J-approximate lower derivatives (Fl ap(xo)) are defined similarly. If Fl ap(XO) = Efap(x0), their common value is called the f-approximate derivative of F at x0 (Fl-ap(Xo)). To prove the above-mentioned result, we need a preliminary lemma and some theorems. Received by the editors January 10, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 26A21, 26A24. ?c1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page 669 This content downloaded from 207.46.13.148 on Sun, 11 Sep 2016 04:20:43 UTC All use subject to http://about.jstor.org/terms