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[SOLVED] Continuous example

Juliayaho

New member
Apr 11, 2013
13
An example of an absolutely continuous f: [0,1] -> ℝ with infinitely many points at which f is not differentiable?

Now what I had in mind was weierstrass function which says that f(x) = Sum (n=0 to infinity) of 1/2^n cos(3^n x) and is continuous everywhere but the derivative exists nowhere...
But I am not sure if that example really fits the criteria of the question or if there might be another more suitable example of that...

Thank for the help.
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,191
Why don't you just use a sine-like function, where you alternate between straight lines of slope $+1$ and straight lines of slope $-1$? You're not required to have the function differentiable nowhere, it seems. You could make this function non-differentiable at every integer. Something like this.

[EDIT]: Oops, I didn't see that the domain has to be $[0,1]$. You could probably take my function and squash it down in a limit, similar to the Weierstrass function.
 
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TheBigBadBen

Active member
May 12, 2013
84
An example of an absolutely continuous f: [0,1] -> ℝ with infinitely many points at which f is not differentiable?

Now what I had in mind was weierstrass function which says that f(x) = Sum (n=0 to infinity) of 1/2^n cos(3^n x) and is continuous everywhere but the derivative exists nowhere...
But I am not sure if that example really fits the criteria of the question or if there might be another more suitable example of that...

Thank for the help.
I'm not sure that the Weierstrass function qualifies as absolutely continuous.

Here's an example that should:

let \(\displaystyle f(x)\) be the piecewise continuous function connecting the points \(\displaystyle \left( \frac{1}{2k+1},0 \right)\) and \(\displaystyle \left( \frac{1}{2k},\frac{1}{2k(k+1)} \right)\) for all integers \(\displaystyle k≥1\).
 
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