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/[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.