On the local fractional derivative of everywhere non-differentiable continuous functions on intervals

• Prove that the local fractional derivative of a nowhere differentiable function on an open interval is not continuous.
• Prove that the nontrivial local fractional derivative does not exist everywhere on an interval.
• Give a criterion of the nonexistence of the local fractional derivative of everywhere non-differentiable continuous functions.
• Construct two everywhere non-differentiable continuous functions on (0,1) and prove that they have also no local fractional derivatives.

We first prove that for a continuous function f(x) defined on an open interval, the Kolvankar-Gangal’s (or equivalently Chen-Yan-Zhang’s) local fractional derivative f(α)(x) is not continuous, and then prove that it is impossible that the KG derivative f(α)(x) exists everywhere on the interval and satisfies f(α)(x) ≠ 0 in the same time. In addition, we give a criterion of the nonexistence of the local fractional derivative of everywhere non-differentiable continuous functions. Furthermore, we construct two simple nowhere differentiable continuous functions on (0, 1) and prove that they have no the local fractional derivatives everywhere.