Condition for a Physically Accepted Wave function

1. $\psi\left(x,t\right)$ should be finite, single-valued and continuous everywhere in space.

2. $\frac{d\psi}{dx}$ should be continuous everywhere in space. But $\frac{d\psi}{dx}$ may be discontinuous in some cases as follows:

(i) If the potential under which the particle is moving, has an infinite amount of discontinuity at some points.

(ii) If the potential under which the particle is moving, is of Dirac delta nature.

3. $\psi\left(x,t\right)$ should be square integrable

i.e.

$\intop_{-\infty}^{\infty}|\psi\left(x,t\right)|^{2}dx$ = finite quantity.