Recall that the Weierstrass function is given by
$f(x) = \sum_{n=0}^{\infty} a^n \cos(b^n\cdot\pi\cdot x)$ where $0, $b$ is a positive odd integer, and $ab>1+\frac{3}{2}\cdot\pi$.
It is well known that this function is continuous everywhere but differentiable nowhere.
My question: is $f$ is uniformly continuous on $\mathbb{R}$? Further, Are there any general techniques to tackle such a problem?
Thanks!
Edit: as noted above, as asked, the question is silly, simply because since $b$ is an integer, the function is periodic, and hence uniformly continuous. But what if we remove the restriction that $b$ is an integer? as far as I know, the function is still continuous everywhere but differentiable nowhere. Is it still uniformly continuous?