I am reading the book by Gitman et al. 'self-adjoint extensions in quantum mechanics'.
In the book, they give a precise definition of the domain of the hamiltonian of an infinite square well.
For me, the point is that, if a wave function belongs to the domain, then itself and its first derivative should be absolutely continuous. This means the tent function below is not in the domain---its first derivative is discontinuous.
This is somehow surprising to me. As I remember, we had excercises in which we were asked to expand the function in terms of the eigenstates of the well. We can then evolve the state in time. Everything seems okay.
So, what is the problem with this state? Why should we rule it out from quantum mechanics?