THE GENERALIZED ZAHORSKI CLASS STRUCTURE OF SYMMETRIC DERIVATIVES

A generalized Zahorski class structure is demonstrated for symmetric derivatives. A monotonicity theorem is proved and a condition sufficient to ensure that a symmetric derivative has the Darboux property is presented. 1. Preliminaries. In 1950 Z. Zahorski (9) began a classification of the derivatives of continuous functions based upon the structure of their associated sets. He defined a descending sequence of subclasses of the Darboux-Baire one functions which form a stratification of the class of derivatives he considered. Kundu (6), in 1976, gener- alized the Zahorski classes in order to extend the theorems of Zahorski to include symmetric derivatives which have the Darboux property and continuous primitives. Our purpose in this work is to replace Kundu's requirement that the primitives be continuous by the more general requirement of measurability and, when possible, to remove the requirement that the symmetric derivatives have the Darboux property. It will be assumed here that, unless it is specifically mentioned otherwise, all functions are finite valued and have their domains contained in R, the real numbers. If f is a function, then we denote C(f) {x:fis continuous at x} and