Say we have a vector function $\vec{D}$ defined in some region on whose boundary its divergence goes to infinity and inside we have $\nabla \cdot \vec{D}=\rho$.
Then is it valid to use the Gauss divergence theorem here? That is can we say :
$\int_{V}(\nabla \cdot \vec{D}) d v=\int_{V} \rho d v$ , hence
$\int_{s}\vec D\cdot d \vec s =\int_{V} \rho d v$
(This is done by in his electrodynamics textbook)
When the divergence of $\vec D$ is infinite on the boundary that doesn't mean that $\vec D$ is infinite too. So the surface integral of $\vec D$ can be performed without being infinite.
So, yes, we can apply the theorem.
If only one partial derivative in the divergence is infinite, the total divergence is infinite too. So we can consider one direction only. Say the r-component (in spherical coordinates). When the derivative at Gauss surface becomes infinite, you can imagine that this doesn't have to be the case for $\vec D$, if $\vec D$ itself. It only means that the tangent (derivative) in the r-direction becomes a vertical line, while $\vec D$ remains finite.
