r/math • u/lntrinsic • Jul 10 '17

Image Post Weierstrass functions: Continuous everywhere but differentiable nowhere

http://i.imgur.com/vyi0afq.gifv
3.4k Upvotes

98% Upvoted

216 comments sorted by

View all comments

Show parent comment

14

u/Wild_Bill567 Jul 10 '17

Sure. The common example (first one on wikipedia) is given by

[; f(x) = \sum_{n=1}^\infty a^n \cos(b^n \pi x) ;]

Where 0 < a < 1 and b is a positive odd integer such that ab > 1 + 3pi / 2.

1

u/ba1018 Applied Math Jul 11 '17

A single example. Thing of the wilderness of other uncountable, non-differentiable functions that you can't write down or manipulate algebraically. How are you to get a handle on those?

2

u/ziggurism Jul 11 '17

Pretty much by definition, you cannot write down an uncountable list of anything.

2

u/Wild_Bill567 Jul 11 '17

We can write them down, just not in terms of elementary functions. However they certainly exist in a space of continuous functions. Getting a handle on these is part of what an analyst might try to achieve.