@IosifPinelis Please see my update.

@IosifPinelis You can download the PDF of the book from here. (i) $\partial \Omega$ is negligible means that $\mu (\partial \Omega) = 0$. (ii) $h:\mathbb R^d\to [0, \infty)$. (iii) This is the essential part of my confusion...

Ah by "corresponding limit", you meant the limit $\lim_{z\to x} \frac{f(z) - f(x)-\langle p, z-x\rangle}{|z-x|}$ I wrote, right?

I'm confused by "corresponding limit does not need to exist" because $\liminf_{z \to x}$ always exists in the extended real numbers. Could you elaborate on this point?

@Gro-Tsen Thank you so much for your help! Could you post your previous comment as an answer?

@Gro-Tsen Could you confirm if my following understanding is correct? $f$ is said to be subdifferentiable at $x$, with subgradient $p$, if there is a continuous function $\omega: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ such that $\omega(r)=o(r)$ as $r \rightarrow 0$, and $$ \forall z \in U; \quad f(z) \geq f(x)+\langle p, z-x\rangle-\omega(|z-x|). $$ Of course, $\omega$ depends on both $x$ and $p$.

@leomonsaingeon I have a question about Theorem 1.17 here. If you don't mind, please have a check in it.

@JHM Some months ago, I encountered the paper Determination of functions by metric slopes which may be of your interest...