Recall that, given an extended real-valued function $f: \mathbb{R}^n \to (-\infty, \infty]$

Its effective domain is, $$\text{dom}(f) = \{x \in \mathbb{R}^n : f(x) < +\infty\}$$

The subdifferential is, $$\partial f(x) = \{v \in \mathbb{R}^n: f(x^\prime) \geq f(x) + v^\top (x^\prime - x), \forall x^\prime \in \mathbb{R}^n\}$$

and its effective domain is all the vectors where it is subdifferentiable,

$$\text{dom}(\partial f) = \{x \in \mathbb{R}^n | \partial f(x) \neq \varnothing\}.$$

Now by a theorem of Rockafellar (top of page 227, "Convex Analysis", 1970), it states,

Let $f: \mathbb{R}^n \to (-\infty, \infty]$ be a closed, proper, convex function, then $$\text{rint}(\text{dom}(f))) \subseteq \text{dom}(\partial f) \subseteq \text{dom}(f).$$ where $\text{rint}$ is the relative interior.

**My question is, when does strict inclusion holds, especially for** $\text{dom}(\partial f) \subseteq \text{dom}(f)$.

For any set I can think of that is closed, you can make a point on the boundary of that set, and there will exist some subgradient at that point. There is the possibility that this strict inequality is an equality for most cases.