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.
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".
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:
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)
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.