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Comment by jhkim | Hacker News Reader

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?

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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.

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.

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".

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