Context
This question is similar, but not identicle, to [3]. Though in [3], it appears the central question revolves around a piece-wise transition between two regions of the real number line.
From [1],
``Suppose $V$ is a subset of $\mathbb{R}^n$, which is compact and has a piece-wise smooth surface $S$. If $\mathbf{F}$ is a continuously differentiable vector field defined on a neighborhood of $V$, then $$\iiint_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)\,dV= \oint_S (\mathbf{F}\cdot\mathbf{\hat{n}})\,dS .''$$
Let's say, for example, that $\mathbf{F}$ is due to a finite countable number of discrete electric charges located inside of $V$. Let's further say that the position of the $i^\text{th}$ electric charge is $\mathbf{r}_i$. Since the electric field is an inverse square-law, the electric field is undefined at positions $\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3,\ldots$. Similarly the derivative does not exist at those positions. Yet, the divergence theorem holds.
Questions
(1) Am I misunderstanding what is meant by ``neighbourhood of a set?'' [2]
(2) Am I erroneous in my understanding that the electric field is not continuously differentiable at the positions $\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3,\ldots$?
(3) Is the predicate ``If $\mathbf{F}$ is a continuously differentiable vector field'' either too restrictive, or inadequately specified? Maybe it should read: If $\mathbf{F}$ is a continuously differentiable vector field at all locations in $V$ up to a countable finite collection of points of measure 0.?
(4) Would a more suitable predicate by, if $\mathbf{F}$ is a continuously differentiable vector field in a neighborhood of $S$?
(5) Bottom line: what am I not understanding here?
Citations
[1] Wikipedia contributors. Divergence theorem [Internet]. Wikipedia, The Free Encyclopedia; 2020 Dec 22, 08:32 UTC [cited 2021 Feb 19]. Available from: https://en.wikipedia.org/w/index.php?title=Divergence_theorem&oldid=995678588.
[2] Wikipedia contributors. Neighbourhood (mathematics) [Internet]. Wikipedia, The Free Encyclopedia; 2021 Feb 12, 00:18 UTC [cited 2021 Feb 19]. Available from: https://en.wikipedia.org/w/index.php?title=Neighbourhood_(mathematics)&oldid=1006271578.