Why is the derivative of $\Psi$ continuous at $x=0$? [duplicate]

I am reading an one-dimensional barrier problem. To evaluate the unknown constants $A,B,C,D$, the continuity of $\Psi(x)$ and $\frac{d\Psi}{dx}$ is used at $x=0$ where the potential has a discontinuity. I understand continuity of $\Psi$. But why is the derivative of $\Psi$ continuous at $x=0$? It has a step function which is not differentiable at $x=0$.

If the potential at some point is continuous or has finite discontinuty then from the Schrodinger equation one can show that the derivative of $\psi$ at that point is continuous.