I Where do discontinuities in the electromagnetic field occur?

[cuallito]

Obviously at point charges, but where along boundaries? Would they theoretically occur in superconductors since they can carry infinite current (J -> infinity)?

 

[vanhees71]

One example are jumps of the normal component of the electric field along conducting surfaces, carrying a surface charge density. The jump is $E_{n1}-E_{n2}=\sigma/\epsilon_0$.

Superconductors must be treated differently. They cannot be described by simply making the resistance 0 (or the electric conductivity to $\infty$). A nice effective theory is the London theory:

https://en.wikipedia.org/wiki/London_equations

 

[tech99]

One example are jumps of the normal component of the electric field along conducting surfaces, carrying a surface charge density. The jump is $E_{n1}-E_{n2}=\sigma/\epsilon_0$.

Superconductors must be treated differently. They cannot be described by simply making the resistance 0 (or the electric conductivity to $\infty$). A nice effective theory is the London theory:

https://en.wikipedia.org/wiki/London_equations

In practice the resistance of the conductor allows the wave to slightly penetrate. I think the problem in general is that the wavelength is finite and so there is no discontinuity at the microscopic level. For instance, total internal reflection in a prism is accompanied by reactive fields in the air behind the reflecting surface.