Discontinuity of a derivative

Question

We know that if $f'(x)$ is discontinuous at $x_{0}$, then $x_{0}$ is a fundamental essential discontinuity of $f'(x)$ (because derivatives can't have removable discontinuity or a jump discontinuity). So why derivative of absolute value has jump discontinuity in $x=0$?

Answer 1

The theorem that you are using is this:

Darboux's theorem: If $I$ is an interval of $\mathbb R$ and $f\colon I\longrightarrow\mathbb R$ is differentiable, then $f'$ has no jump discontinuities.

You cannot apply it to the absolute value function and to the point $0$ since that function is not differentiable at $0$.