Counterexamples on Jumarie's three basic fractional calculus formulae for non-differentiable continuous functions

TL;DR

This paper presents five counterexamples demonstrating that Jumarie's three basic fractional calculus formulae are invalid for non-differentiable continuous functions, challenging their general applicability.

Contribution

The paper provides the first explicit counterexamples to Jumarie's fractional calculus formulae for non-differentiable functions, highlighting their limitations.

Findings

Jumarie's formulae do not hold for non-differentiable continuous functions.

Five explicit counterexamples are constructed to demonstrate the failure.

The results question the validity of Jumarie's fractional calculus in certain contexts.

Abstract

In the present paper, we provide five counterexamples to show directly the Jumarie's formulae are not true for non-differentiable continuous functions.