Even though the field becomes infinite near the charge a gaussian surface can pass through the continuous charge distribution. How?

Question

Even though the field becomes infinite near the charge a gaussian surface can pass through the continuous charge distribution and have a finite value of flux. How?

Answer 1

In the case of a continuous distribution of sources, one computes the flux by breaking up the distribution into a arbitrarily large number of arbitrarily small sources.

So first, you now have an arbitrarily small amount of charge, generating an incredibly small net flux through your entire surface. Moreover, there is only an arbitrarily small part of your surface that is very close to your terribly small source. In the limit (which means one needs to invoke calculus to justify this properly), the flux due to your very small source very close to a very small patch of your surface actually contributes only a very small amount, although when you sum the contributions of all the small sources over all the small patches the result is actually a finite flux.

Answer 2

Maxwell's equations refer to charge density and current density, not to point charges. They do indeed stop making sense for point charges because it would take infinite energy to confine a charge at a point.