Nowhere-differentiability of Weierstrass' function and cases of slow frequency growth

The nowhere-differentiability of Weierstrass' classical function is given a short proof with a few basics from integration theory. In terms of the Fourier transformation, the method is (a simple example of) a second microlocalisation. It also yields nowhere-differentiability of functions with almost quadratic frequency growth. An article is to appear in J. Fourier Analysis and Applications.