I've a vector function whose curl is well defined inside and outside,except on the boundary
$\mathbf{\nabla} \times \mathbf{H}=\mathbf{J}_{f}$
I'm interested in using stokes theorem across the boundary (i.e the area on which I apply the theorem resides both inside and outside the boundary ) and I'll get $$\oint \mathbf{H} \cdot d \mathbf{l}=I_{f_{\mathrm{enc}}}$$
Will the closed loop integral be valid that'll result from the application of stokes theorem, notwithstanding that the function from which it was derieved had a discontinuity at the boundary?