I recently stumbled on the Weierstrass function, whose main claim to fame (as I understand it) is to be continuous everywhere, but non-differentiable everywhere as well. Apparently I was in good company with Gauss' and others who assumed that to be impossible! I guess I'm asking, is this dichotomy mostly down to the loosened definition of continuity, i.e. Hölder continuous instead of Lipschitz continuous? Not that I dispute the validity or use of the former definition, but it certainly would mean that my intuition was correct in the stricter Lipschitz sense.