A function has a critical point if its derivative fails to exist at the point or if its derivative is zero at that point.

If a function $\displaystyle f$ is not continuous at $\displaystyle x_0$ then it has no derivative at $\displaystyle x_0$ so that is a critical value.

On the other hand, the slope on the left at $\displaystyle x_0$ may not be the slope on the right at $\displaystyle x_0$, in that case we call that a "spike". There is no derivative there.