Discontinuities in fields can be offset by nonunique arrangements of singular sources and fields at the interface. In the recent past, it has been shown that discontinuity in the transverse electric field does not necessarily involve a magnetic current distribution. The boundary condition can be formulated in a number of alternative ways without affecting the solution outside the interface. Surface electric and magnetic current distributions used to compensate for the discontinuity in electric tangential field are interchangeable, leading to a variety of boundary conditions. This concept is further developed here to present a comprehensive description for all field and source components. In the absence of magnetic sources, a singular magnetic field component and a highly singular current source are needed. No singular electric field or highly singular charges are present Boundary conditions are derived for all field discontinuities. In particular, a boundary condition is worked out for a discontinuous normal magnetic field without using magnetic charges. It is shown that this discontinuity is offset by the normal component of the transverse curl of the highly singular electric current source. All results are shown to be consistent with a generalized continuity equation.