Is the derivative of $|x|$ defined for every x in the neighborhood?

Question

Is the derivative of $f(x)= |x|$ defined for every $x$ in the neighborhood?

I know the derivative of $x=0^+$ and $x=0^-$ is not the same. But is it correct to say that the derivative is defined for every $x$ in the neighborhood of $0$?

Answer 1

No, it is not correct, since $0$ belongs to every neighborhood of $0$, and $f$ is not differentiable at $0$.