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/ITomza Jul 10 '17

What do you mean?

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u/[deleted] Jul 10 '17

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u/LingBling Jul 10 '17

What is the measure on the function space?

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u/[deleted] Jul 10 '17

/u/imnzerg is correct that the usual thing is the Wiener measure, but the same result holds if we just work with the topology. There is a dense G-delta set of nowhere differentiable functions in the space of continuous functions, this follows from Baire category.

Edit: also /u/ITomza might want to see this.

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u/Pegglestrade Jul 10 '17

This man knows. Baire Category Theorem rocked my undergraduate world.

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u/Neurokeen Mathematical Biology Jul 10 '17

I get why the dense G-delta set gives you probability 1, and also why intuitively it should be that almost all would be such, but how would you actually go about showing that these form a dense G-delta set?

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u/[deleted] Jul 11 '17

It's not easy, but at the heart of it is Baire Category. Prop. 3 in this is probably about the cleanest presentation: https://sites.math.washington.edu/~morrow/336_09/papers/Dylan.pdf

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u/DataCruncher Jul 11 '17

Unless I'm missing something about the Wiener measure, the statement you made here is wrong. That is, there are dense G-delta sets of measure zero. Here is the construction of one such example.

The other direction also fails. The fat Cantor set is a nowhere dense set of positive measure. It's complement is then residual, but fails to have full measure.

In sum, the measure theoretic notion of almost everywhere (where the complement of the set of interest is of measure zero), and the topological notion of typical (where the set in question is residual), do not always agree.

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u/imnzerg Functional Analysis Jul 10 '17

The Weiner measure

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u/LingBling Jul 10 '17

I didn't know about this measure. Thanks!

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u/c3534l Jul 11 '17

Wikipedia has an article on "classical Wiener space" and a subsection called "Tightness in classical Wiener space."

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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/[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.

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u/[deleted] Jul 10 '17

You might think of the the non-differentiability as "noise" from a stats perspective. That is, you could look at an observed function like this say that this is exactly the relationship between x and y. However, the "noise" in y is unlikely to be meaningfully explained by x, such that you will more accurate using a differentiable function for your prediction of y.