I'm reading about *subdifferentiable function* at page 232 of Villani's *Optimal Transport: Old and New*.




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**Definition 10.5** (Subdifferentiability, superdifferentiability). Let $U$ be an open set of $\mathbb{R}^n$, and $f: U \rightarrow \mathbb{R}$ a function. Then:


* (i) $f$ is said to be *subdifferentiable* at $x$, with subgradient $p$, if
$$
f(z) \geq f(x)+\langle p, z-x\rangle+o(|z-x|) .
$$
The convex set of all subgradients $p$ at $x$ will be denoted by $\nabla^{-} f(x)$.
* (ii) $f$ is said to be *uniformly subdifferentiable* in $U$ if there is a continuous function $\omega: \mathbb{R}\_{+} \rightarrow \mathbb{R}\_{+}$, such that $\omega(r)=o(r)$ as $r \rightarrow 0$, and
$$
\forall x \in U \quad \exists p \in \mathbb{R}^n ; \quad f(z) \geq f(x)+\langle p, z-x\rangle-\omega(|z-x|).
$$
* Corresponding notions of *superdifferentiability* and *supergradients*
are obtained in an obvious way by just reversing the signs of the inequalities.




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**My understanding** I think $o(|z-x|)$ is the Landau symbol, i.e., $g (x) = o(|z-x|)$ means that $\lim\_{z \to x} \frac{g(x)}{|z-x|} = 0$.



> 
> Could you explain what $f(z) \geq f(x)+\langle p, z-x\rangle+o(|z-x|)$ means?
> 
> 
> 


I'm not sure if it means that the limit
$$
\lim\_{z\to x} \frac{f(z) - f(x)-\langle p, z-x\rangle}{|z-x|}
$$
exists and is non-negative.



What does Landau symbol mean in an inequality?
Landau $o(\cdot)$ notations should be interpreted in inequalities as inferior/superior limits.
In this case in particular, it $f(z)\ge f(x)+⟨p,z−x⟩+o(|z−x|)$ is equivalent to
$$
 \liminf\_{z \to x} \frac{f (z) - f (x) - \langle p, x - z\rangle}{\vert z - x\vert} \ge 0;
$$
the corresponding limit does not need to exist.


