The nondifferentiability of Weierstrass' function (Q1198799)

In 1872, Weierstrass proved that the function \[ f(x)=\sum^ \infty_{n=0}a^ n\cos(b^ n\pi x), \] where \(b\) is an odd integer, \(0<a<1\), and \(ab>1+3\pi/2\), is not differentiable for any value of \(x\). Weierstrass' result has since been generalized rather widely by a number of authors, who considered functions of the general forms: \[ C(x)=\sum^ \infty_{n=0}a_ n\cos(b_ nx)\quad\text{ and } S(x)=\sum^ \infty_{n=0}a_ n\sin(b_ nx) \] where the \(a\)'s and \(b\)'s are positive, the series \(\sum a_ n\) is convergent, and the \(b\)'s increase steadily and with more than a certain rapidity. Following the monumental work on this subject by \textit{G. H. Hardy} [Trans. Am. Math. Soc. 17, 301- 325 (1916)], the present authors set \(a_ n=a^ n\) and \(b_ n=b^ n\), and discuss improvements upon some of the nondifferentiability conditions for \(C(x)\) and \(S(x)\) given by Hardy himself.