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Consider the electric field due to volume charge distribution in volume $V'$:

$\mathbf{E}=\displaystyle \int_{V'} \rho' \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV'$

The integrand blows up at a field point $P \in V'$. But the integral remains finite. This can be shown by choosing $P$ as the center of our Cartesian coordinate system and switching to spherical coordinate system.

Now consider the electric field due to surface charge distribution in surface $S'$:

$\mathbf{E}=\displaystyle \int_{S'} \sigma' \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dS'$

At a field point $P \in S'$, it can be shown using Gaussian pillbox argument that there is a discontinuity of $4\pi \sigma'$ in the field $\mathbf{E}$ and hence the field is undefined at a point $P \in S'$.

Now as $P$ gets closer and closer to the surface, the integrand blows up. Now how shall we show that the integral doesn't blow up (or approaches a limit)?

Conditions:

  1. $\sigma'$ is nowhere infinite
  2. $S'$ is a closed and bounded surface (finite surface)
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  • $\begingroup$ Rephrasing your question: are you trying to prove that a function that is measurable almost everywhere can still have a well defined surface integral? If so, this is a well studied field of complex analysis and residuals theorems and the answer lies in there (totally independent of this application to the electric field). $\endgroup$ – gented May 29 '19 at 11:31
  • $\begingroup$ @gented: Then how did people (eg: Poisson, Maxwell,etc) dealt with it before the birth of complex analysis? $\endgroup$ – N.G.Tyson May 29 '19 at 11:52
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    $\begingroup$ Do you mean how they dealt with proving that it still converges or how they calculated it? For the former I don't know for sure but I believe they just didn't care :)...for the latter you can just practically integrate it by using the definition of surface integral and Dini's theorem (which reduces everything to standard Riemann integrals). $\endgroup$ – gented May 29 '19 at 11:55
  • $\begingroup$ I see... So could it be that most of electromagnetism became rigorous after the introduction of these advanced topics (after Maxwell's death)? $\endgroup$ – N.G.Tyson May 29 '19 at 12:02
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    $\begingroup$ Relevant: physics.stackexchange.com/q/450426 $\endgroup$ – jacob1729 May 29 '19 at 16:28

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