OK, what calculus knowledge I have left me utterly and completely baffled by the Weierstrass function.
Everywhere continuous but nowhere differentiable?? That really seems so intuitively impossible. Not that I’m doubting a constructionist, but I couldn’t actually find how the nowhere-diff is proved in my googling. I found one online text that said it would “prove it later” but I never found the proof.
Can anyone explain the proof generally or link to an explicit proof? What a strange function…
I know nothing about this but was bored and thought I’d take a stab at a search.
I’m thinking this site might be your guy who didn’t finish the proof.
Have you tried the Weierstrass Institute for Applied Analysis and Stochastics? If anyone has it they probably would. I started searching around but quickly became lost in the myriad responses. You might have better luck knowing exactly what you’re looking for.
This website shows you an approximate idea of what the function looks like; you can kind of think of the function as having a “corner” at every real number:
http://www.mathworld.com/WeierstrassFunction.html
I have a rigorous proof of this around my apartment somewhere, but I’m out of town right now (and I don’t want to try and cook up a proof from scratch). If you really want to see it, and no one else posts one in the meantime, I can post one in a week or so.
Yes and no, actually. It’s strange because it’s not what we’re used to seeing. However, the Weierstrass function is not really an oddity at all, in a certain sense: most of the continuous functions from the reals to reals are differentiable nowhere. It’s the ones that we usually see, and differentiate, that are odd.
FWIW, the function is referred to as the Weierstrass-Mandelbrot function over at mathworld. That might help in searching. I’ll poke around and see if I can find anything.
[naive question]
Is the path of a particle undergoing Brownian motion everywhere continuous and nowhere differentiable? It’s permanence in 3-space guarantees continuity (I think) and it’s random motion implies that there is never any well-defined velocity, i.e. that it’s nowhere differentiable.
[/naive question]
BTW, the Koch snowflake may be a better example of a continuous everywhere, differentiable nowhere curve that you can think of somewhat intuitively.
http://www.mathworld.com/KochSnowflake.html
KarlGauss, I’m afraid I don’t know enough about Brownian motion in general to answer your question definitively, but I’ve been under the impression that the two ideas are at least very similar.
Yes, although the best reason I can give is that one of my professors said so.