Composite Continuous Path Systems and Differentiation

Abstract

The concept of composite differentiation was introduced by O'Malley and Weil to generalize approximate differentiation. The concept of continuous path systems was introduced by us. This paper combines these concepts to introduce the notion of composite continuous path systems into differentiation theory. It is shown that a number of results that hold for composite differentiation and for continuous path differentiation also hold for composite continuous path differentiation. In particular, a composite continuous path derivative of a continuous function is a Baire class one function on some dense open set, and extreme composite continuous path derivatives of a continuous function are Baire class two functions. It is also shown that extreme composite continuous path derivatives of a Borel measurable function are Lebesgue measurable. Finally, for each composite continuous path system $E$, continuous functions typically do not have $E-$derived numbers with $E-$index less than one.