I'm working on mathematically proving the Gauss' Law, and I'm having proving the approximations of the electric field integral converge uniformly. The steps are:
First I take discrete charges $q_i$ at locations $a_i\in \mathbb{R}^3$. The electric field is $$E(x)=\underset{i=1}{\overset{k}{\sum}}\frac{1}{4\pi\varepsilon}q_{i}\frac{\underline{x}-\underline{a}_{i}}{||\underline{x}-\underline{a}_{i}||^{3}}$$ where $\varepsilon$ is the vacuum permittivity. For an open set $U\subset\mathbb{R}^3$ if $\partial U$ is a smooth surface taking the surface integral of the second kind yields: $$\iint_{\partial U}E\cdot\hat{n}dS=\underset{\{i;q_{i}\in U\}}{\sum}\frac{q_i}{\varepsilon}$$
Next, we look at a charge density $\rho(x)$, continuous on the closure of an open set $U\subset\mathbb{R}^3$. The electric field now takes an integral form:$$E(x)=\frac{1}{4\pi\varepsilon}\int_{U}\frac{x-y}{||x-y||^{3}}\rho(y)dy$$ I use approximations for the field - Taking cubes of sidelength $s$ centered at $s\mathbb{Z}^3$ we consider the cubes $\{C_i\}$ fully contained in $U$, and approximate the integrand by it's value at the centre point $\frac{x-a_{i}}{||x-a_{i}||^{3}}\rho(a_{i})$. I am able to show that we have pointwise convergence: $$E(x)=\lim_{s\rightarrow0}\underset{i}{\sum}\frac{1}{4\pi}\int_{C_{i}}\frac{x-a_{i}}{||x-a_{i}||^{3}}\rho(a_{i})dy=\frac{1}{4\pi}\cdot\lim_{s\rightarrow0}\underset{i}{\underset{i}{\sum}\frac{x-a_{i}}{||x-a_{i}||^{3}}\rho(a_{i})Vol(C_{i})}$$ I want to use the first result, and push the surface integral through the electric field integral to get: $$\iint_{\partial U}E\cdot\hat{n}dS=\frac{1}{4\pi}\cdot\underset{i}{\sum}4\pi\rho(a_{i})Vol(C_{i})=\lim_{s\rightarrow0}\underset{i}{\sum}\rho(a_{i})Vol(C_{i})=\frac{1}{\epsilon}\iiint_{U}\rho$$ $$\rightarrow \iint_{\partial U}E\cdot\hat{n}dS=\frac{Q_{in}}{\varepsilon}$$ which is Gauss' Law.
The problem is I can't seem to be able to prove uniform convergence of these approximations. The integrand is not continuous and goes to $\infty$ when $x=y$, and I don't know how to work it to get uniform convergence. Any help will be greatly appriciated!