Proof why wave function is continuous

Question

Is there an accurate proof why wave function is continuous? I mean wave function as coefficient of eigen states when representing a state with eigen states, so, I am asking continuity of $\psi(x)=<\psi|x>$.
I understand that it is a function of $\mathcal{L}^2$, but how does it lead to $\psi(x)$ being of $C^2$?

Answer 1

Not a proof, no. But a reason.

The momentum and energy operators are derivatives of $\psi$. As $\psi$ approaches discontinuous, the derivatives get large, and expectation values of $p$ and $E$ approach infinity.

The form of the momentum operator follows from momentum being the generator of translation. See https://en.wikipedia.org/wiki/Momentum_operator

The form of the Energy operator can be derived from the fact that $\psi$ is a wave function. See https://en.wikipedia.org/wiki/Energy_operator.