Symmetric Differentiation on Time Scales

TL;DR

This paper introduces a symmetric derivative for functions on arbitrary time scales, expanding the differentiability framework to include functions not differentiable in the traditional delta or nabla sense.

Contribution

The paper defines a new symmetric derivative on time scales and demonstrates its applicability to functions outside the scope of existing derivatives.

Findings

Symmetric derivative exists on various time scales.

Functions not delta or nabla differentiable can be symmetric differentiable.

Properties of the symmetric derivative are established.

Abstract

We define a symmetric derivative on an arbitrary nonempty closed subset of the real numbers and derive some of its properties. It is shown that real-valued functions defined on time scales that are neither delta nor nabla differentiable can be symmetric differentiable.