Continuity of convolution

Question

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be continuous and bounded function, $\phi: \mathbb{R}\rightarrow \mathbb{R}$ be in $C^\infty$. Is it true that $F(x) = \int_{-\infty}^\infty \phi(x-t)f(t)$ is in $C^\infty$? And how to prove it if it's true?