Peano path derivatives

In this paper we introduce Peano path derivatives as a natural extension of the notion of path derivatives. We give a sufficient condition on a system of paths to ensure the corresponding Peano path derivative is Baire 1. As consequences, we obtain that unilateral approximate and unilateral -approximate Peano derivatives are Baire one.

     

A mean value theorem for generalized Riemann derivatives

Functional differences that lead to generalized Riemann derivatives were studied by Ash and Jones in (1987). They gave a partial answer as to when these differences satisfy an analog of the Mean Value Theorem. Here we give a complete classification.

     

Infinite approximate Peano derivatives

In this paper we introduce approximate Peano derivatives with infinite values allowed, and we show that these derivatives are Baire one, and possess the Darboux and Denjoy-Clarkson properties. Also we show that if they are bounded from above or below on an interval, then the corresponding ordinary derivatives exist and equal the approximate Peano derivatives.

     

Extending n times differentiable functions of several variables

It is shown that n times Peano differentiable functions defined on a closed subset of and satisfying a certain condition on that set can be extended to n times Peano differentiable functions defined on if and only if the nth order Peano derivatives are Baire class one functions.