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Comment by Bromskloss | Hacker News Reader

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

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This page has the "zooming in" animation and a good explanation:

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).

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?

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.

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-...)

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".

From the way they try to explain it, I think they meant to say a "countable" number of points, instead of finite.

On what basis are you calling this assertion "clearly not true"?

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

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)

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.

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

Thanks. This actually sums it better than the article.

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.

It talks about Koch's snowflake and Brownian motion.

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