How to compute the Clarke subdifferential of the function $h(x,y)=x^2+y^2$?

I've been reading an article on Clarke critical values of subanalytic Lipschitz functions. There I've come across the following definition(s) of subdifferential: $$f: U \to \mathbb{R}^n, \ \ \ \ \emptyset \neq U \subset \mathbb{R}^n$$ $f$ is locally Lipshitz continuous Let $$x \in U:$$ The Fre...

In the book [1], Clarke et al. define the generalized gradient for a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ as follows. 8.1. Theorem (Generalized Gradient Formula). Let $x\in\mathbb{R}^n$, and let $f:\mathbb{R}^n\to\mathbb{R}$ be Lipschitz near $x$. Let $\Omega$ be any subset of zer...

According to various websites, for some function $f:X\to R$ we can define a map $$ D(x,v):= \lim_{y\to x} \sup_{t\searrow0} \frac{f(y+tv)-f(y)}{t} $$ and the Clarke Subdifferential (Def 1) is $$ \partial f(x) := \{v \in X^* : D(x,v) \geq v\}. $$ The Clarke Subdifferential generalizes the gra...

The Clarke subdifferential of $f(x)=|x|$ at $x=0$ is a set $[-1, 1]$, and $\{sign(x)\}$ otherwise, just like the ordinary subdifferential because of convex. Now for \begin{equation} g(x)= \begin{cases} |x|/2-x^2/4, \text { if }|x|\leq 1,\\ 1/4, \text { otherwise}, \end{cases} \end{equa...

To give some context: I am aware of the uses of Convex Analysis (and its applications in Convex Optimization), I have been studying (for a while) the developments of Nonsmooth Analysis (and its applications in Nonsmooth Optimization) as traced by Frank Clarke. While all of this work on subdiffe...