$f(x) \in AC(\mathbb{R}) \cap L^1(\mathbb{R})$ such that $f'(x) \notin L^1(\mathbb{R})$

Asked Aug 26 '19 at 13:12

Active Aug 26 '19 at 13:12

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I have to find some examples of functions $f(x) \in AC(\mathbb{R}) \cap L^1(\mathbb{R})$ such that $f'(x) \notin L^1(\mathbb{R})$.

I thought about it but can't find any example. Where can I direct my line of reasoning in order to solve this type of questions?

Can anyone help me?

asked Aug 26 '19 at 13:12

Phi_24

1

Maybe $\frac{\sin{e^{x^4}}}{x^2+1}$? The idea is to find a smooth $L^1$ function whose derivative is very much unbounded, so you want it to oscillate very fast. – Aphelli Aug 26 '19 at 13:17
@Mindlack but is this function absolutely continuous? – Phi_24 Aug 26 '19 at 13:31
@Phi_24 If $f$ is continuously differentiable then $f\in AC$. – David C. Ullrich Aug 26 '19 at 13:41
@Mindlack You can give a much simpler example, like $\frac{\sin(x^2)}{x^2+1}$. – David C. Ullrich Aug 26 '19 at 13:43
How do you define $AC(\mathbb R)?$ – zhw. Aug 26 '19 at 14:01
https://en.m.wikipedia.org/wiki/Absolute_continuity – Aphelli Aug 26 '19 at 14:13
@Mindlack https://math.stackexchange.com/questions/353473/differentiable-but-not-absolutely-continuous?rq=1 – Phi_24 Aug 26 '19 at 15:49

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