Does the Weierstrass function have a point of increase? - MathOverflow

Does the Weierstrass function have a point of increase?

2011-10-04T12:49:32Z

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.

Answer by GH for Does the Weierstrass function have a point of increase?

2011-10-05T04:06:59Z

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.

Answer by Gerry Myerson for Does the Weierstrass function have a point of increase?

2011-10-05T05:14:07Z

A similar function is proved to be nowhere monotonic in Gelbaum and Olmsted, Counterexamples in Analysis, Chapter 2, Example 21.