This is the entire text of the problem.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that shows it is false.
$$\text{If } div \mathbf{F} = 0 \text{ then }\iint _S \mathbf{F}\cdot d \mathbf{S}= 0 \text{ for every closed surface S.}$$
The textbook says the answer is $\mathbf{true}$. This is a Calculus III, Multivariable Calculus text so it's possible to prove without sophisticated real analysis.
How do we know the partial derivatives are continuous? Yes, the divergence is zero but I can create two functions whose sum is zero but have discontinuities. Because it's not usually covered until Analysis, the text does not mention the following theorem. "A function with a finite [or can they be countably finite?] number of discontinuities is integrable." So of course a function exists whose derivative has any number of discontinuities. Let this function be P_x, Q = -P and R =$0$. Then $\mathbf{F} = Px + Qy + Rz$ has divergence = $0$. I can justify the partials being finite. I see two options.
A. Does Divergence theorem apply even when the partial derivatives are discontinuous?
B. Is there some way to prove from $div\mathbf{F}=0$, that the partial derivatives are continuous?
This is from $\mathbf{\text{Calculus Early Transcendentals}}$ by Soo T. Tan, copyright 2011, section 14.8, page 1307, question number 27.
