The Weierstrass function's' odd qualities

  • I
  • Thread starter rumborak
  • Start date
  • #1
706
154
I recently stumbled on the Weierstrass function, whose main claim to fame (as I understand it) is to be continuous everywhere, but non-differentiable everywhere as well. Apparently I was in good company with Gauss' and others who assumed that to be impossible!

I guess I'm asking, is this dichotomy mostly down to the loosened definition of continuity, i.e. Hölder continuous instead of Lipschitz continuous? Not that I dispute the validity or use of the former definition, but it certainly would mean that my intuition was correct in the stricter Lipschitz sense.
 

Answers and Replies

  • #2
.Scott
Homework Helper
2,894
1,156
I recently stumbled on the Weierstrass function, whose main claim to fame (as I understand it) is to be continuous everywhere, but non-differentiable everywhere as well. Apparently I was in good company with Gauss' and others who assumed that to be impossible!

I guess I'm asking, is this dichotomy mostly down to the loosened definition of continuity, i.e. Hölder continuous instead of Lipschitz continuous? Not that I dispute the validity or use of the former definition, but it certainly would mean that my intuition was correct in the stricter Lipschitz sense.
No, it is continuous in the true sense of the term. But what is d(abs(x))/dx when x=0? The problem with the Weierstrass function is that it has that problem for all x.
 
  • #3
35,725
12,313
According to this discussion, it is Hölder continuous for every ##0<\alpha<1##, but not Lipschitz continuous (which would be ##\alpha=1##).
These are stricter criteria than the regular continuity, however. The function is continuous.
 
  • Like
Likes S.G. Janssens
  • #4
706
154
No, it is continuous in the true sense of the term. But what is d(abs(x))/dx when x=0? The problem with the Weierstrass function is that it has that problem for all x.

Oooh, that is an excellent explanation, thanks a lot. It's been a long time since college math, I had forgotten that a "kink" in a graph is non-differentiable. From there it's not too hard to construct an "only kinks" function.
 
  • #5
S.G. Janssens
Science Advisor
Education Advisor
991
774
I guess I'm asking, is this dichotomy mostly down to the loosened definition of continuity, i.e. Hölder continuous instead of Lipschitz continuous? Not that I dispute the validity or use of the former definition, but it certainly would mean that my intuition was correct in the stricter Lipschitz sense.
Yes, indeed. Even more: By Rademacher's theorem, every Lipschitz continuous function is differentiable almost everywhere, which in a sense is the opposite of being nowhere differentiable.
From there it's not too hard to construct an "only kinks" function.
It is more difficult than it seems, since the "kinks" have to be everywhere.

In addition to the references given in the link in the post by @mfb , there is also a pair of "Insights" written about it, but they seem to be missing a third part and I have not read them myself. Maybe @jbunniii would like to comment.
 

Related Threads on The Weierstrass function's' odd qualities

  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
15
Views
4K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
6K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
9
Views
7K
Top