Simple proofs of nowhere-differentiability for Weierstrass's function and cases of slow growth (Q2380387)

It is known that the Weierstrass function, with real parameters greater than \(1\), is continuous but nowhere differentiable [\textit{G. F. Hardy}, American M. S. Trans. 17, 301--325 (1916; JFM 46.0401.03)]. In the present paper, another proof of this fact is given. In fact, a more general continuous function is considered and nowhere differentiability is proved in two results, under different conditions. Such results are presented and described in a very elucidative way and their conclusions also hold for the real and imaginary parts of \(f:\mathbb R\to\mathbb C\), where \(f(t)=\sum_{j=0}^\infty a_j\exp (ib_jt)\), with \(\sum_{j=0}^\infty |a_j|<\infty\) and \(0<b_j\uparrow\infty\). Conditions for nowhere Lipschitz continuity of \(f\) are also investigated. The author also makes comments on the Hölder continuity of order \(\alpha\in\,]0,1[\) of \(f\). Finally, some examples are presented to illustrate the results.