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The monster in question is the Weierstraß function [0], noteworthy for being continuous everywhere yet differentiable nowhere. I don't if the article talks ab

The monster in question is the Weierstraß function [0], noteworthy for being continuous everywhere yet differentiable nowhere. I don't if the article talks about other "monsters" too. I didn't look further than to this one. [0] https://en.wikipedia.org/wiki/Weierstrass_function https://en.wikipedia.org/wiki/Weierstrass_function

jhkim 9y ago
It seems like the equation shown in the article doesn't satisfy the condition listed on the wikipedia page. Did the author make a mistake?
- supergarfield 9y ago
  The Wikipedia article gives Weierstrass's original condition, but it was later improved to just ab ≥ 1 (as noted, not very visibly, at the end of the article). However, the Nautilus article also says that "Conventional wisdom held that for any continuous curve, it was possible to find the gradient at all but a finite number of points", which is clearly not true, so I'd be cautious about its technical correctness.
  - euyyn 9y ago
    From the way they try to explain it, I think they meant to say a "countable" number of points, instead of finite.
  - alimw 9y ago
    On what basis are you calling this assertion "clearly not true"?
    - bonzini 9y ago
      You can define a periodic function that goes from 0 to 1 as a straight line and then from 1 to 0 as another straight line, basically a triangle wave. It is not differentiable on an infinite (and countable) number of points: https://en.m.wikipedia.org/wiki/Triangle_wave https://en.m.wikipedia.org/wiki/Triangle_wave
      - lmm 9y ago
        I imagine they would have been talking about functions on the interval rather than our modern sense of functions on the reals. (Of course there are plenty of examples on the interval as well, but no obvious ones)
        bonzini 9y ago
        Take something like arcsin sin pi/2/x(x-1), where "arcsin sin pi*x/2" is "pointy" like a triangle wave. There you have it.
  - vilhelm_s 9y ago
    There is a lot of discussion at this Stack Exchange question, "Is Kline right that Cauchy believed that continuous functions must be differentiable?" (https://hsm.stackexchange.com/questions/3480/is-kline-right-that-cauchy-believed-that-continuous-functions-must-be-differenti https://hsm.stackexchange.com/questions/3480/is-kline-right-...) Mathematicians didn't state this as theorem or axiom, so it's hard to pin down exactly what they might have believed; the people answering the question talk about "except at isolated points" rather than "except at a finite number of points".
bonzini 9y ago
It talks about Koch's snowflake and Brownian motion.
dorfsmay 9y ago
Thanks. This actually sums it better than the article.
acqq 9y ago
This page has the "zooming in" animation and a good explanation: https://sites.math.washington.edu/~conroy/general/weierstrass/weier.htm https://sites.math.washington.edu/~conroy/general/weierstras... It's mentioned in the Wikipedia article: "The Weierstrass function could perhaps be described as one of the very first fractals studied, although this term was not used until much later. The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line." Does anybody know any even earlier "fractal studied" before they were named so? I guess the author probably meant "Bolzano function" (~1830, published in 1922, mentioned in the Notes section).
Asooka 9y ago
It can also be integrated, and can be used as the basis for doing Fourier transforms. Completely useless as far as I've researched, but curious nonetheless. Here's an example paper: https://arxiv.org/abs/1502.07734 https://arxiv.org/abs/1502.07734
phkahler 9y ago
I take issue with this. The function is an infinite series of cosine functions which can be shown to be convergent everywhere. It's also easy to write a series for the derivative of it, but that is probably divergent and hence not well defined anywhere. It's a nice trick. But if one truncates the series at any finite number of terms, the problem goes away and it is differentiable everywhere. I see parallels to the countability of the reals, and a similar issue with vague notions of infinity.

jhkim 9y ago

It seems like the equation shown in the article doesn't satisfy the condition listed on the wikipedia page. Did the author make a mistake?

supergarfield 9y ago
The Wikipedia article gives Weierstrass's original condition, but it was later improved to just ab ≥ 1 (as noted, not very visibly, at the end of the article). However, the Nautilus article also says that "Conventional wisdom held that for any continuous curve, it was possible to find the gradient at all but a finite number of points", which is clearly not true, so I'd be cautious about its technical correctness.
- euyyn 9y ago
  From the way they try to explain it, I think they meant to say a "countable" number of points, instead of finite.
- alimw 9y ago
  On what basis are you calling this assertion "clearly not true"?
  - bonzini 9y ago
    You can define a periodic function that goes from 0 to 1 as a straight line and then from 1 to 0 as another straight line, basically a triangle wave. It is not differentiable on an infinite (and countable) number of points: https://en.m.wikipedia.org/wiki/Triangle_wave https://en.m.wikipedia.org/wiki/Triangle_wave
    - lmm 9y ago
      I imagine they would have been talking about functions on the interval rather than our modern sense of functions on the reals. (Of course there are plenty of examples on the interval as well, but no obvious ones)
      - bonzini 9y ago
        Take something like arcsin sin pi/2/x(x-1), where "arcsin sin pi*x/2" is "pointy" like a triangle wave. There you have it.
- vilhelm_s 9y ago
  There is a lot of discussion at this Stack Exchange question, "Is Kline right that Cauchy believed that continuous functions must be differentiable?" (https://hsm.stackexchange.com/questions/3480/is-kline-right-that-cauchy-believed-that-continuous-functions-must-be-differenti https://hsm.stackexchange.com/questions/3480/is-kline-right-...) Mathematicians didn't state this as theorem or axiom, so it's hard to pin down exactly what they might have believed; the people answering the question talk about "except at isolated points" rather than "except at a finite number of points".

bonzini 9y ago

It talks about Koch's snowflake and Brownian motion.

dorfsmay 9y ago

Thanks. This actually sums it better than the article.

acqq 9y ago

This page has the "zooming in" animation and a good explanation: https://sites.math.washington.edu/~conroy/general/weierstrass/weier.htm https://sites.math.washington.edu/~conroy/general/weierstras... It's mentioned in the Wikipedia article: "The Weierstrass function could perhaps be described as one of the very first fractals studied, although this term was not used until much later. The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line." Does anybody know any even earlier "fractal studied" before they were named so? I guess the author probably meant "Bolzano function" (~1830, published in 1922, mentioned in the Notes section).

Asooka 9y ago

It can also be integrated, and can be used as the basis for doing Fourier transforms. Completely useless as far as I've researched, but curious nonetheless. Here's an example paper: https://arxiv.org/abs/1502.07734 https://arxiv.org/abs/1502.07734

phkahler 9y ago

I take issue with this. The function is an infinite series of cosine functions which can be shown to be convergent everywhere. It's also easy to write a series for the derivative of it, but that is probably divergent and hence not well defined anywhere. It's a nice trick. But if one truncates the series at any finite number of terms, the problem goes away and it is differentiable everywhere. I see parallels to the countability of the reals, and a similar issue with vague notions of infinity.