Let $f, \phi:\mathbb{R} \rightarrow \mathbb{R}$ let $f \in C^k$, $\phi \in C_0^\infty$ and $\phi \geq 0$, $\int \phi(x)dx=1$.
Why a function $g(x,t)=\int f(x-ty)\phi(y)dy$ for $t > 0$, $x \in \mathbb{R}$, is of class $C^\infty$.
It probably follows from the following fact, however I don't know in such a way, that a convolution $(u *\phi)(x)=\int u(x-y)\phi(y)dy$, $x \in \mathbb{R}$, of every locally integrated function $u: \mathbb{R} \rightarrow \mathbb{R}$ (i.e integrated on every compact) with function $\phi \in C_0^\infty$, is a smooth function and $(u *\phi)^{(n)}=u*\phi^{(n)}$ for every $n \in \mathbb{N}$.
Thanks.