Example: A common problem in undergraduate electromagnetic classes is calculating the electric field inside a solid spherical charge distribution with known charge density. The common method of solution in my experience is as follows: 1) Place the origin of a coordinate system at the center of the sphere. 2) Using symmetry and a spherical Gaussian surface, calculate E with Gauss' Law. Here's the problem... The most IMO rigorous derivation I have seen of Gauss' Law seems to be valid only if there is no charge on the boundary of the Gaussian surface. This made sense to me, because if you calculate the electric field using the "brute-force" method of integration using Coulomb's Law, I don't believe the field would be defined at any point inside the charge distribution. For example, when using this brute-force method, the denominator of the integrand has a (r - r') term in it. Since the integration is with respect to the source point r', it seems that all calculations of this form must implicitly assume that the field point r under consideration cannot be an element of the volume being integrated over (if it was, there would be division by zero). So, I am wondering if anyone can explain why Gauss' Law is valid inside volume charge distributions and why the electric field (or potential for that matter)is defined inside volume charge distributions. I am mostly looking for a mathematical reason, because this is really a math problem. I know that IRL there is separation between charges, etc. but we are modeling the charge distribution as a continuum in this type of problem. Hope my question makes sense.