**Problem**

The Weierstrass function $W(x)$ is given by

$W(x)=\sum_{n\geq 0} a^n \cos(b^n \pi x)$

where $0< a <1$ and $b$ is an odd integer such that $ab > 1+3\pi/2$.

A function $f:\mathbb{R}\rightarrow \mathbb{R}$ is said to have a point of increase if there exists a $t \in \mathbb{R}$ and $\delta>0$ such that

$f(t-s)\leq f(t) \leq f(t+s) \quad \forall s \in [0,\delta]$.

So my question is does the Weierstrass function have a point of increase?

**Motivation**

In Burdzy's paper there is a proof that a Brownian motion does not have a point of increase. There are examples of nowhere differentiable functions which have a point of increase that one could construct but I have been having difficulty seeing if the Weierstrass function does.

I would be grateful for any references or heuristics regarding this problem, or any comments as to the difficulty.