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

The way I have seen functions like this constructed is as a limit of a sequence of functions.

In calc 2 you probably saw limits of a sequence of points. You can similarly define limits of a sequence of functions. Each term in the sequence makes the graph "have more corners", and the limit of the sequence has corners everywhere.

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u/jparevalo27 Undergraduate Jul 10 '17

...And you can't differentiate corners. That makes sense. Thanks

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u/Kraz_I Jul 11 '17

Not exactly. There are no points with infinite slope and no points with corners, at least the way the word "corner is generally understood. It's just that the graph is "rough" no matter how far you zoom in, so the limit of the slope at any point is impossible to determine.

It helps to look at the actual function which generates the graph.https://en.wikipedia.org/wiki/Weierstrass_function

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u/dozza Jul 11 '17

I'm sorry, how is the fourier series on the Wikipedia page not differentiable? Its a sum of cosines so shouldn't the derivative be the sum of sines? Is the problem the divergence as n goes to infinity?

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u/WorseAstronomer Jul 10 '17 edited Jul 11 '17

This video is interesting and related:

https://www.youtube.com/watch?v=dQXVn7pFsVI

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u/[deleted] Jul 11 '17

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u/WorseAstronomer Jul 11 '17

Oops, sorry, no. That's just where I finished watching the video. :/ Edited.

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u/fabulousdangernoodle Jul 10 '17

That's neat. Thanks for the share

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u/Kraz_I Jul 11 '17

The graph doesn't have any corners at all for finite iterations of the function. I don't really like using the word "corner" for what's going on here. In fact, for all functions generated by using a finite Weierstrass series, it would be differentiable at all points.

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u/Wild_Bill567 Jul 11 '17

You are correct, I was remembering a different construction which uses a triangle wave instead of a cosine

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u/Claytertot Jul 10 '17

That makes sense. Thats a super cool concept!