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.