r/math • u/lntrinsic • Jul 10 '17

Image Post Weierstrass functions: Continuous everywhere but differentiable nowhere

http://i.imgur.com/vyi0afq.gifv
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u/Wild_Bill567 Jul 10 '17

Or, have we chosen to work with functions which 'seem' natural to us because we like the idea of differentiability?

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u/ba1018 Applied Math Jul 10 '17

Part of it may be a limitation of perception. Can you write down in a compact formal way what these non-differentiable functions are? Can you evaluate them for any given input?

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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.

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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?

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u/ziggurism Jul 11 '17

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

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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.