**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](https://digital.lib.washington.edu/dspace/bitstream/handle/1773/2168/paper27.pdf?sequence=1) 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.



Does the Weierstrass function have a point of increase?
The original proof of Weierstrass (see pages 4 to 7 in Elgar (ed.): Classics on Fractals, Westview Press, 2004) constructs, for any $x\_0\in\mathbb{R}$, two sequences $(x'\_n)$ and $(x''\_n)$ such that 
$$x'\_n < x\_0 < x''\_n,\qquad x'\_n\to x\_0,\qquad x''\_n\to x\_0,$$ 
but
$$\frac{W(x'\_n)-W(x)}{x'\_n-x}\qquad\text{and}\qquad
\frac{W(x''\_n)-W(x)}{x''\_n-x}$$
are of opposite signs and their absolute values tend to infinity. This shows that $W(x)$ has no point of increase and no point of decrease.


