# Continuity of electric field

I was solving a problem in which i came across a radially symmetric discontinuous electric field, which puzzled me for a moment but then I figured that there must be surface charge density at the point of discontinuity causing this.

My question is :

Is such discontinuity in 3D always the result of a singularity like a surface charge distribution, line charge or point charge? By singularity I mean infinite volume charge density. If so, why?

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 $\epsilon_\circ \Delta E$.
$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$