Weierstrass nowhere differentiable function

The Weierstrass function is $$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^n\pi x),$$ where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$.

Properties

Theorem: The Weierstrass function $f$ is continuous everywhere but differentiable nowhere.

Proof: █