Consider the following functions, associated with certain trigonometrical sums: $$ f_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty}\frac{\cos(n^{\alpha+\beta}x)}{n^{\alpha}},\qquad g_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty}\frac{\sin(n^{\alpha+\beta}x)}{n^{\alpha}}.$$ For $\alpha>1$, they clearly belong to $C^{0}(\mathbb{R})$ since they are total converging series of continuous functions. My questions now are:
- Is is true that $f_{\alpha,\beta}$ and $g_{\alpha,\beta}$ are nowhere-differentiable for any $\alpha>1,\beta\geq 0$?
- For which values of $\alpha,\beta$ is $f_{\alpha,\beta}$ continuous and nowhere-differentiable?
- For which values of $\alpha,\beta$ is $g_{\alpha,\beta}$ continuous and nowhere-differentiable?