Yes, a discontinuity in the electric field is always associated with a charge distribution with infinite density. That's a necessary implication of Gauss' law. To think of it qualitatively, where there is a discontinuity in the field, there must be field lines either starting or ending. (If it's not obvious why that is, think about the fact that the electric field magnitude is different on one side of the discontinuity than on the other. There will be more field lines on the side with the stronger side, or else lines pointing in different directions on the different sides if the amplitude is positive on one side and negative on the other.) But electric field lines start and end only on charges, or at infinity.
To think about it even more physically, consider the force on a positive test charge. It will be different on either side of the discontinuity. But electric forces are exerted by charges, so the only way to produce a jump in the force across the discontinuity is to have some charge there.
In fact, it turns out that if the electric field amplitude jumps by $\Delta E$ across a discontinuity, there must be a surface charge there (charge per unit area) of magnitude $\epsilon_\circ \Delta E$.