Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions

The well-known Banach-Mazur theorem says that every separable Banach space can be isometrically embedded into $C([ 0,1])$. We prove that this embedding can have the property that the image of each nonzero element is a nowhere approximatively differentiable and nowhere Hölder function. It improves a recent result of L. Rodriguez-Piazza where the images are nowhere differentiable functions.