The Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. The function is described as follows:
\begin{equation}\label{eq:weierstrass_func} f(x)=\sum_{n=0}^{\infty} a^n \cos \left(b^n \pi x\right) \end{equation}
where $0<a<1$, $b>1$ an integer, and
$$ a b>1. $$
I would like to show that the Weierstrass function is a Hölder function with exponent $\frac{log 1/a}{log b}$.
I have already seen this answer link. But in fact I don't understand the line: $$ \text { Since } b>1 \text { and } h \in(-1,1) \text { there exist } p \in \mathbb{N} \text { such that } 2^{-1} b^p|h| \leq 1<b^p|h| $$ If $b$ and $h$ are already fix I dont see a way to choose $p$ so that $2^{-1} b^p|h| \leq 1$.
Moreover why we restrict only to the case $h \in(-1,1)$? The Hölder condition should holds on all of $\mathbb{R}$.
Additionaly I found this Wikipedia link. Here there is a hint, the function is written equivalently as $$ W_\alpha(x)=\sum_{n=0}^{\infty} b^{-n \alpha} \cos \left(b^n \pi x\right) $$ using the substitution $\alpha=-\frac{\ln (a)}{\ln (b)}$. Then $W_{\mathrm{a}}(x)$ is Hölder continuous of exponent $\alpha$ ( there exists $C$ so that $$ \left|W_\alpha(x)-W_\alpha(y)\right| \leq C|x-y|^\alpha $$ for all $x$ and $y$).
Can someone please explain to me with more details? Thank you!
