In my Quantum mechanics 1 lecture the professor proofed that the wave function in one dimension has to be continuous as long as the potential is "well behaved". My question is whether the wave function still has to be continuous for an arbitrary potential, which is arbitrarily "wild". Do you know a proof that it has to be continuous/does not have to be continuous or some arguments why it should/should not be the case?
I understand that from a physical standpoint, the wavefunction must be continuous, because else we will get an infinite kinetic energy, which is unphysical. But in physics generally all potentials are continuous as well when you look at the smallest dimensions.
But I still pose this question because I think it is somewhat different from the other questions which have been asked on this topic. What this question is about is the purely mathematical point of view, if we take the whole physics out of it. If one assumes there can be arbitrarily "wild" potentials (for example a potential which has a delta function at x if x is rational and is zero if x is irrational), does the wave function still have to be continuous?
