Does the Weierstrass function have a point of increase? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T15:18:42Z http://mathoverflow.net/feeds/question/77128 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77128/does-the-weierstrass-function-have-a-point-of-increase Does the Weierstrass function have a point of increase? Bati 2011-10-04T12:49:32Z 2011-10-05T05:14:07Z <p><strong>Problem</strong></p> <p>The Weierstrass function $W(x)$ is given by</p> <p>$W(x)=\sum_{n\geq 0} a^n \cos(b^n \pi x)$</p> <p>where $0&lt; a &lt;1$ and $b$ is an odd integer such that $ab > 1+3\pi/2$.</p> <p>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</p> <p>$f(t-s)\leq f(t) \leq f(t+s) \quad \forall s \in [0,\delta]$.</p> <p>So my question is does the Weierstrass function have a point of increase?</p> <p><strong>Motivation</strong></p> <p>In <a href="https://digital.lib.washington.edu/dspace/bitstream/handle/1773/2168/paper27.pdf?sequence=1" rel="nofollow">Burdzy's paper</a> 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.</p> <p>I would be grateful for any references or heuristics regarding this problem, or any comments as to the difficulty.</p> http://mathoverflow.net/questions/77128/does-the-weierstrass-function-have-a-point-of-increase/77200#77200 Answer by GH for Does the Weierstrass function have a point of increase? GH 2011-10-05T04:06:59Z 2011-10-05T04:06:59Z <p>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 $$x'_n &lt; x_0 &lt; x''_n,\qquad x'_n\to x_0,\qquad x''_n\to x_0,$$ 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.</p> http://mathoverflow.net/questions/77128/does-the-weierstrass-function-have-a-point-of-increase/77202#77202 Answer by Gerry Myerson for Does the Weierstrass function have a point of increase? Gerry Myerson 2011-10-05T05:14:07Z 2011-10-05T05:14:07Z <p>A similar function is proved to be nowhere monotonic in Gelbaum and Olmsted, Counterexamples in Analysis, Chapter 2, Example 21. </p>