Convolution of a Continuous Function with a Compactly Supported Continuous Kernel

Let $n\ge1$ be a natural number, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^n$, let $d$ denote the Euclidean distance, a metric on each Euclidean space, and let $\lambda_n$ be Lebesgue measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$. Write $\bar B(x,r)$ for the closed ball in $(\mathbb{R}^n,d)$.

Let $\Omega\subseteq\mathbb{R}^n$ be open in $(\mathbb{R}^n,d)$, let $f:\Omega\to\mathbb{R}$ be continuous on $\Omega$ as a map from $(\mathbb{R}^n,d)$ to $(\mathbb{R},d)$, let $\delta>0$ be a real number, and let $\rho:\mathbb{R}^n\to\mathbb{R}$ be continuous on $\mathbb{R}^n$ with $\rho(y)=0$ for every $y$ with $\lVert y\rVert>\delta$. Let

$$\Omega^{\delta}=\{x\in\mathbb{R}^n:\bar B(x,\delta)\subseteq\Omega\},$$

the set shown to be open in The $\delta$-Interior of an Open Subset of $\mathbb{R}^n$ is Open, and for $x\in\Omega^{\delta}$ let $h_x:\mathbb{R}^n\to\mathbb{R}$ be given by $h_x(y)=f(x-y)\rho(y)$ for those $y$ with $x-y\in\Omega$ and $h_x(y)=0$ for all other $y$, the difference $x-y$ being that of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$.

The convolution of $f$ with $\rho$ is the function $f*\rho:\Omega^{\delta}\to\mathbb{R}$ given by

$$(f*\rho)(x)=\int_{\mathbb{R}^n}h_x\,d\lambda_n\qquad(x\in\Omega^{\delta}),$$

the integral being that of Integrable Function and the Lebesgue Integral; it is a real number, since $h_x$ is integrable by claim 1 of The Convolution Integrand is Continuous, Compactly Supported and Integrable.