The Convolution of a Continuous Function with a Continuous Kernel is Continuous

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, and let $d$ denote the Euclidean distance.

Then $f*\rho$ is continuous on $\Omega^{\delta}$, as a map from $(\mathbb{R}^n,d)$ to $(\mathbb{R},d)$.