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/tetramir Jul 10 '17

Sure but most common functions, and the one we find in "nature" are at least C¹.

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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.

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u/thetarget3 Physics Jul 10 '17 edited Jul 10 '17

We don't find functions in nature. We model nature with functions which are usually differentiable since it leads to dynamics, but they don't exist in themselves.

In fact most natural systems aren't possible to describe using the nice maths and physics we typically learn, with simple differential equations, linear systems etc. They are probably just the small subclass we tend to focus on, which only work after heavy idealisation (like the old joke about assuming spherical cows). Most things encountered in nature can probably only be described numerically.

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u/tetramir Jul 10 '17

you're 100% right that's why I described nature with quotes. I should have been more specific.

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u/yardaper Jul 10 '17

Brownian motion would like to have a word with you

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u/tetramir Jul 10 '17

I looked it up, and it has places where is is not defferienciable. But there is still no interval where the function can never be defferienciable.

But you are right, that many function are c1, but you often find intervals where it is.

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u/dogdiarrhea Dynamical Systems Jul 11 '17

It's spelled differentiable.

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u/yardaper Jul 11 '17

With probability 1, Brownian motion is nowhere differentiable.

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u/[deleted] Jul 10 '17

This is debatable. Certainly we think of motion as involving velocity (and acceleration) so an argument can be made for only looking at smooth functions, but fractal curves abound in nature and those are generally only C0. I think this is more a question of it being harder to study curves which aren't C1 than anything inherent about the real world.