Function $f\in L^1(\mathbb{R})$ which is absolutely continous but $f'\notin L^1(\mathbb{R})$

Question

I need to find a function $f$ with the properties required in the title, so $f\in AC(\mathbb{R})\cap L^1(\mathbb{R})$ such that $f'\notin L^1(\mathbb{R})$, where $AC(\mathbb{R})$ denotes absolutely continous functions on the $\mathbb{R}$.

I've been told that absolutely continous functions are those for which $f'\in L^1_{loc}(\mathbb{R})$, so intuition suggests this $f$ I'm looking for needs to be "small" at infinity but also oscillating enough so that its derivative isn't globally integrable.

Thanks in advance for your help.

Answer 1

Just an idea, an analytic function in $L^1(\mathbb{R})$ with derivative not in $L^1(\mathbb{R})$ would work. How about $$f(x) = \frac{\sin x^2}{x^2+1}\\ f'(x) = \frac{2 x \cos x^2}{x^2+1} - \frac{\sin x^2} {(x^2 + 1)^2}$$