Consider a continuous function $f:[0,\infty) \rightarrow \mathbb{R}$ such that $\lim_{x\rightarrow\infty} \int_{0}^{x}f(x)dx = 0$.
I would like to know if under this condition we have $\lim_{x\rightarrow\infty} \int_{0}^{x}f^2(x)dx = 0$.
Any help is appreciated.