Given a current-carrying surface, it makes sense to ask what the component $B_\perp$ of the magnetic field is perpendicular to the surface, which is $B_\perp = \BB\cdot\nn$, where $\nn$ is the unit normal to the surface. The component parallel to the surface, $B_{\parallel}$, is more subtle, since there are an infinite number of directions parallel to the surface. However, since for a current-carrying surface there is a preferred direction in the surface, namely the direction of the current $\KK$, we can distinguish between the component in the surface and parallel to the current, called $B_{\parallel\parallel}$, and the component in the surface but perpendicular to the current called $B_{\parallel\perp}$.

Using an infinitesimally small Gaussian surface and an infinitesimally small Amperian loop, you should find how these different components of the magnetic field change from one side of a surface current to the other. Are they continuous? If not, can you express the discontinuity in terms of physical properties of the surface current?