The classification of generalized Riemann derivatives

TL;DR

This paper characterizes when one generalized Riemann derivative implies another and classifies all such derivatives into equivalence classes, providing a comprehensive understanding of their relationships.

Contribution

It provides a complete characterization of pairs of generalized Riemann derivatives where differentiability in one implies differentiability in the other, and classifies all such derivatives into equivalence classes.

Findings

Identified conditions under which one generalized Riemann derivative implies another.

Established the equivalence relation among generalized Riemann derivatives.

Determined the structure of equivalence classes of these derivatives.

Abstract

We characterize all pairs $(\mathcal{A}$,$\mathcal{B})$ of generalized Riemann differences for which $\mathcal{A}$-differentiability implies $\mathcal{B}$-differentiability. Two generalized Riemann derivatives $\mathcal{A}$ and $\mathcal{B}$ are equivalent if a function has a derivative in the sense of $\mathcal{A}$ at a real number $x$ if and only if it has a derivative in the sense of $\mathcal{B}$ at $x$. We determine the equivalence classes for this equivalence relation.