How to find the subdifferential of $|x|$?

Question

I want to compute the subdifferential of $ f $ on $ \mathbb{R} \setminus \{ 0 \} $ when $ f(x) = |x| $. How do I do this?

Answer 1

On $\mathbb{R}^*$, $f$ is differentiable so $\partial f(x)=\{f'(x)\}$ with $f'(x)$ the differential of $f$ at $x$.

You can also see Subdifferential of a finite dimensional function.