Is speed of light continuous when entering a medium? I know that light has the speed $c$ in vacuum and speed $c/n$ in a medium with refractive index $n$.
I wonder how this exactly happens - is there some kind of smooth transition? If so, on which scale? Or is there some kind of discontinuity?
 A: To understand this, we need to clarify a few things. This is a more QM answer.
Photons always travel at speed c in vacuum when measured locally.
Photons always travel in vacuum, even in the medium, between the atoms.
Now how does then the speed of light get slower then c in the medium?
It is because we are talking about the speed of the wavefront. The individual photons do travel at speed c in the medium, between atoms. Now the trick is, that the individual photons do not travel in a straight line in the medium. As they interact with the medium's atoms, they get scattered and change angle. At each interaction with each atom they change angle. Thus, they move in a zig zag motion through the medium.
Now when you try to calculate the speed of light, you use a straight path length, and you divide it with the time it takes for the light to get through the medium. What you actually measure as time, is how long it takes for the wavefront to get through the medium.
Now since the path of the individual photons is not straight, but zig zag, they in reality travel a longer path then the straight one that you use for your calculation. So you will get a speed slower then c, because the actual photons making up the wavefront, will travel a longer path, and it takes them more time to get through the medium.
Now you are asking whether the speed of light changes smoothly or with discontinuity.
The answer is that it changes with discontinuity. It is because as the photons reach the medium, they start interacting with the atoms and get scattered and change angle, and start moving in a zig zag way.
So the wavefront moves at speed c in vacuum, and as soon as the wavefront reaches the medium, it moves slower at its speed that is determined by the medium. There is no transitional speed inbetween.
A: Say sunlight from the sun hit the surface of a freshwater lake. The index of refraction for air is around 1, and the index of refraction for water is 1.33. Since n changes right as the sunlight hits the water, this means that c/n changes right as it hits the water as well. Thus, it is a discontinuity.
