We know the fact that $\partial_{\bar z} (1/z) \propto \delta(z, \bar z)$. But I am slightly confused about the proof. I have seen a few of them, and of course they use the Green's theorem, or the Stokes' theorem,
$$\int_D \partial_{\bar z}\frac{ 1 }{ z } dz\wedge d\bar z \sim \int_{\partial D} \frac{ 1 }{ z } dz \ .$$
But if I understand correctly, these two theorems requires the integrand $1/z$ on the right to have continuous derivative, or roughly speaking the integrand on the left should have some sort of continuous property.
So I wonder why these two theorems still hold/are effective in this case? Do the two theorems admit generalization to handle some singularities like this one?