19th century mathematicians: all curves that are continuous are differentiable

weierstrass: *defines a curve that is totally continuous but non-differentiable at not only a point, but every single point in its existence*

19th century mathematicians: :surprised_pikachu:

@starkatt @dankwraith Oh yeah, those pathological little critters! It's bizarre to think that in some sense almost every function is everywhere discontinuous, and almost every continuous function is nowhere differentiable.

@CoronaCoreanici @dankwraith everywhere discontinuous is a concept that's kinda freaking me out, not gonna lie.

@starkatt @dankwraith An entirely fair reaction! I wasn't kidding when I said that mathematicians stare directly into the abyss and take notes.

There's a handful of things like that, little infohazard traps early in math study. The empty set, Thomae's function, the topologist's sine curve. More I forget right now.

Still decidedly worth studying the language Nature speaks.

@CoronaCoreanici @dankwraith heh, I had encountered the topologist's sine curve but didn't know it had a name!

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