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

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((alpha))(x) is not continuous, and then prove that it is impossible that the KG derivative f ((alpha))(x) exists everywhere on the interval and satisfies f((alpha))(x) not equal 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. (C) 2016 Elsevier B.V. All rights reserved.