Consider Schrödinger's *time-independent* equation
$$
-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi.
$$
In typical examples, the potential $V(x)$ has discontinuities, called *potential jumps*.

Outside these discontinuities of the potential, the wave function is required to be twice differentiable in order to solve Schrödinger's equation.

In order to control what happens at the discontinuities of $V$ the following assumption seems to be standard (see, for instance, Keith Hannabus' *An Introduction to Quantum Theory*):

Assumption: The wave function and its derivative are continuous at a potential jump.

**Questions**:

1) Why is it necessary for a (physically meaningful) solution to fulfill this condition?

2) Why is it, on the other hand, okay to abandon twofold differentiability?

Edit: One thing that just became clear to me is that the above assumption garanties for a well-defined probability/particle current.