Help with a problem about the convergence of an improper integral

Question

Let $\int_{0}^1 f$ and $\int_{0}^1 g$ two convergent improper integrals. Is necessary that $\int_{0}^1 f \cdot g$ converge?

Answer 1

Take $f(x)=g(x)=\frac{1}{\sqrt x},\ when \ x\not=0$ and $f(0)=g(0)=0$. Then $fg(x)=\frac{1}{x},\ when\ x\not=0$ and $fg(0)=0$