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

Abstract

Juamrie proposed a modified Riemann-Liouville derivative definition and gave three so-called basic fractional calculus formulae (u(t)and v are required to be non-differentiable and continuous for the first formula, f is assumed to be differentiable for the second formula, while in the third formula f is non-differentiable and u is differentiable, at the point t. I once gave three counterexamples to show that Jumarie's three formulae are not true for differentiable functions(Cheng-shi Liu. Counterexamples on Jumaries two basic fractional calculus formulae. Communications in Nonlinear Science and Numerical Simulation, 2015, 22(1): 92-94.). However, these examples cannot directly become the suitable counterexamples for the case of non-differentiable continuous functions. In the present paper, I first provide five counterexamples to show directly the Jumarie's formulae are also not true for non-differentiable continuous functions. Then I prove that essentially non-differentiable cases can be transformed to the differentiable cases. Therefore, those counterexamples in the above paper are indirectly right. In summary, the Jumarie's formulae are not true. This paper can be considered as the corrigendum and supplement to the above paper.

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