Convolution with a $C^k$ Kernel is of Class $C^k$

Let $n$, $\Omega$, $f$, $\delta$, $\rho$, the set $\Omega^{\delta}$ and the convolution $f*\rho$ be as in Convolution of a Continuous Function with a Compactly Supported Continuous Kernel. By claim 2 of Differentiating a Convolution through the Kernel the set $\Omega^{\delta}$ is open in $\mathbb{R}^n$.

1. (Finite order) Let $k$ be a natural number and suppose that $\rho$ is of class $C^k$ on $\mathbb{R}^n$. Then $f*\rho$ is of class $C^k$ on $\Omega^{\delta}$.

2. (Smoothness) If $\rho$ is smooth on $\mathbb{R}^n$, then $f*\rho$ is smooth on $\Omega^{\delta}$.