Can $\int_0^1 f^2(x)dx<\infty$ imply $\int_0^1 |f(x)|dx<\infty$, if $f$ unbounded near $0$.
Let $g$ be a function from $(0, 1)$ to $\mathbb{R}$ and $\int_0^1 g^2(x)dx$ finite. Does this imply that $\int_0^1 g(x)dx$ is finite?