electromagnetism - What is the tangential component of an electric field is continuous across the boundary between two media?

I see the mathematical derivation for the fact that the tangential component of an electric field across two media is continuous, but I don't intuitively understand how this is the case. The electric field should either be impeded or not depending on the material, and this "impedance" for the electric field should affect tangential components as well.

Perhaps, I am misunderstanding something, but I can't intuitively see how this happens.

Answer

When you have two dielectric materials with homogeneous $\epsilon_1 , \epsilon_2$ each, applying an external electric field will only produce a bounded charge distribution only on the boundary - the interface between the media.

So locally, you have a plane with charge distribution - thus creating an electric field that is perpendicular to the interface, meaning that the perpendicular electric field changed and has a discontinuity, but the parallel has no reason to change, meaning it is still continuous.