Is relative boundary of the epigraph the same as relative interior of the domain of a convex function?

Question

Let $f \colon \mathbb{R^n} \rightarrow \mathbb{R}$ be convex.

From what I understand, a subdifferential $$ \partial f(x) :=\{z:f(x)\,\,\ge\,\,f(x')+ \langle z, x-x' \rangle \ \forall x' \in \text{dom}(f) \} $$ of $f$ at $x$ is guaranteed to be non-empty when $x$ is in the relative interior of the domain of $f$, $\text{dom}(f)$.

I can understand that the subdifferential at $x$ is guaranteed to be nonempty when $x$ is on the relative boundary of the epigraph of $f$. But, I do not understand how it is possible for us to guarantee that the subdifferential will be non-empty on the relative interior.

Are both of these statements true?

1. the subdifferential is always nonempty on the relative interior of the domain.

2. the subdifferential is always nonempty for the points $x \in \mathbb{R^n}$ for which $(x,f(x))$ is a point on the relative boundary of the epigraph.

Answer 1

1) Yes - see Rockafellar's Convex Analysis.

2) No. Consider $f(x)=-\sqrt{x}$ if $x\geq 0$ and $+\infty$ if $x<0$. Then $(0,0)$ is in the (relative) boundary of the epigraph but $\partial f(0)=\varnothing$.