Boundary condition for dielectric to dielectric
Something has to modify the normal component of the normal part of the wave. The discontinuous definition comes from the material. The wave propagates until it sees material, this material looks like a discontinuity to the wave (a step function in material properties if you think about it). 

In addition the normal component is discontinuous if there is a surface charge density, if there is no surface charge then there is no discontinuity:


Source: http://www.antenna-theory.com/tutorial/electromagnetics/electric-field-boundary-conditions.php