Subgradients of non-convex functions

In these notes (section 2.3), it is stated that:

A point $x^*$ is a minimizer of a function $f$ (not necessarily convex) if and only if $f$ is subdifferentiable at $x^*$ and $0 \in\partial f(x^*).$

Could anybody provide me with references for a proof of the above statement?

Is there a reference where we can learn more about subgradients of non-convex functions? In Section 3. Calculus of subgradients of the above notes,many properties of subgradients are presented for convex functions. I would like to know which ones among these properties still hold for non-convex functions.

Thank you in advance!

Solution 1:

$$x^* \text{ minimizes } f(x) \Longleftrightarrow 0\in\partial f(x^*)$$ is trivial by definition of subgradients: $$f(x) - f(x^*) \ge \partial f(x^*)^T (x-x^*)\quad \forall x.$$

Thanks to Michael Grant for his comments.