A number of physics texts I've encountered recently have rather cavalierly applied variants of the Stokes' Theorem on fields with singularities in order to derive various properties of the electromagnetic field.
Griffiths and Jackson both compute the integral of the electric field over a closed surface surrounding a charge directly:
$$\int_S f^*\omega=\int_{S^2}\omega=\frac{1}{4\pi\epsilon_0}\oint_S \frac{\rho(\mathbf{x'})}{(\mathbf{x-x'})^2}\,(\mathbf{x-x'})^2\sin\theta\, d\theta\, d\phi=4\pi$$
and then claim that the integral of $\nabla\cdot\mathbf{E}$ over the region bounded by the surface must also equal $4\pi$ per the divergence theorem. They even justify the $\nabla^2(1/r)=4\pi\delta$ identity using the same argument.
However, every statement and derivation of the divergence theorem or the GST I have seen has required the $k$-form to be at least $C^1$, while this form is not even continuous.
Jackson goes to great lengths in a related problem (showing that the potential integral obeys Poisson's equation) to avoid this problem by constructing an $\eta$-potential
$$V=\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\mathbf{x}')}{\sqrt{(\mathbf{x-x'})^2+\eta^2}}\,d^3x$$
which resembles the actual potential as $\eta$ approaches zero, and then computing the Laplacian directly while expanding the charge density in a Taylor series.
So I'm not at all sure what the differentiability requirements are for the use of Stokes' theorem, and how to demonstrate integrability for a particular form. Are there general classes of discontinuous forms for which Stokes' theorem is valid, or does the validity need to be demonstrated via limits on a case by case basis?