Sequential derivatives

TL;DR

This paper introduces sequential secant and chord derivatives for functions not differentiable at a point, using sequences to analyze their behavior, with special results for the Weierstrass function.

Contribution

It develops a new framework for defining derivatives via sequences, extending classical concepts to non-differentiable functions.

Findings

Chord derivatives at any point of the Weierstrass function cover the entire extended real line.

Sequential derivatives can be characterized for functions with complex local behavior.

The approach generalizes classical derivatives using sequence-based definitions.

Abstract

Consider a real valued function defined, but not differentiable at some point. We use sequences approaching the point of interest to define and study sequential concepts of secant and cord derivatives of the function at the point of interest. If the function is the celebrated Weierstrass function, it follows from some of our results that the set cord derivatives at any point coincides with the extended real line.