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?


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.

  • $\begingroup$ I would say the thing to do is consult Hardy's paper ... Hardy, G. H. (1916) "Weierstrass's nondifferentiable function," Transactions of the American Mathematical Society, vol. 17, pages 301–325. $\endgroup$ Oct 4 '11 at 14:59

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.

  • 1
    $\begingroup$ Note that $W$ can have a local maximum (at $0$, for example) or local minimum. And thus one-sided points of increase or decrease. $\endgroup$ Oct 5 '11 at 11:44
  • $\begingroup$ Unfortunately I can't get a preview of the book but this seems like a nice way to prove the statement. Thanks. $\endgroup$
    – Bati
    Oct 5 '11 at 12:11
  • $\begingroup$ @Gerald: I agree, and I believe it is difficult to find the local minima of $W(x)$. At any rate, Weierstrass' proof gives the following: (1) if $\lfloor 1/2-b^n x_0 \rfloor$ is even infinitely often, then the function is not locally decreasing at $x_0$ from the left and not locally increasing at $x_0$ from the right, in particular $x_0$ is not a local minimum; (2) if $\lfloor 1/2-b^n x_0 \rfloor$ is odd infinitely often, then the function is not locally increasing at $x_0$ from the left and not locally decreasing at $x_0$ from the right, in particular $x_0$ is not a local maximum. $\endgroup$
    – GH from MO
    Oct 5 '11 at 14:26

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

  • $\begingroup$ Actually nowhere monotonic is a weaker property (to me) than having no point of increase or decrease. $\endgroup$
    – GH from MO
    Oct 5 '11 at 5:49
  • $\begingroup$ Nowhere monotonic (ie, no interval on which it is monotonic) is weaker than nowhere differentiable. $\endgroup$ Oct 5 '11 at 8:37
  • 1
    $\begingroup$ Point taken. But Bati still might get something out of Example 21, and, anyway, I never pass up a chance to promote the Gelbaum and Olmsted book. $\endgroup$ Oct 5 '11 at 22:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.