Is there an unbounded integrable function with integrable derivative in $(0,1)$?

Question

I wonder if there is a differentiable unbounded function $f\in L^1(0,1)$ with $f'\in L^1(0,1)$. The elementary examples $x^\alpha$ or $\log x$ suggest that my question should be answered negatively. However, I have no idea how to prove that such a function can not exist.

Answer 1

Let's work on $(-1,1)$ for simplicity. Then (see Real and Complex Analysis (Rudin), Theorem 7.21): $$f(x)=f(0)+\int_0^x f'(x)\,dx,$$ so it can't get too far from $f(0)$ since $f'$ is Lebesgue integrable.