quantum mechanics - Justification for the matching condition for the wave function at potential jumps. Why is it both restrictive enough and sufficiently general?

To answer your first question:

Actually the assumption is not that the wave function and its derivative are continuous. That follows from the Schrödinger equation once you make the assumption that the probability amplitude $langle psi|psirangle$ remains finite. That is the physical assumption. This is discussed in Chapter 1 of the first volume of Quantum mechanics by Cohen-Tannoudji, Diu and Laloe, for example. (Google books only has the second volume in English, it seems.)

More generally, you may have potentials which are distributional, in which case the wave function may still be continuous, but not even once-differentiable.

To answer your second question:

Once you deduce that the wave function is continuous, the equation itself tells you that the wave function cannot be twice differentiable, since the second derivative is given in terms of the potential, and this is not continuous.