Hello,

I would like to ask the following question:

How can I prove that the continuous function given as

$\displaystyle f(x)=\sum_{k=1}^{\infty} \frac{\sin{(kx)}}{k^{p}}$

is not differentiable at any real x?

there is given $\displaystyle p>1$ ..

I thought I could try for arbitrary, but fixed $\displaystyle x \in \mathbb{R}$ to find the sequence $\displaystyle \{x_n\}_{n=1}^{\infty} \subset \mathbb{R} $

such that $\displaystyle x_n>x\,,\,\forall n\in\mathbb{N}$ and $\displaystyle \lim_{n\to\infty}x_n=x $ and for which there holds

$\displaystyle \lim_{n\to \infty}\frac{f(x_n)-f(x)}{x_n-x}=\pm\infty $

but it looks I am not able to do it ..

thank you very much for any ideas