Let $g:\mathbb{R}\longrightarrow\mathbb{R}$ be a locally integrable function and $\phi\in\mathcal{C}^\infty_c(\mathbb{R})$ (i.e. $\phi $is a $\mathcal{C}^\infty$ function with compact support in $\mathbb{R}$)
Is it true that $g\phi$ is integrable, in the sense of Riemann?