Can you explain why do we need the wave function $\Psi$ to be continuous at every point in two ways - An intuitive physical and a rigorous mathematical?
The exact problem I'm confused about is in the problem with a particle in $1$D box with infinite potential walls. I somewhat understand why the wave function has to be zero outside the bounds but why does it need to be zero at them? Is there a deeper meaning than just preserving the continuity of $\Psi$?
