When explaining the continuity of convex functions, it says
Let $f: E \rightarrow (-\infty, +\infty)$ be convex. Then $f$ is continuous on the interior of its domain and, in fact, for any $x \in int\ dom(f)$, there exists an open neighborhood $U$ of $x$ and $L>0$ such that for any $u,v\in U$, $f(u) - f(v) \le L ||u-v||_2$
Initially I thought $int\ dom(f)$ was the epigraph of $f$ but an epigraph isn't exactly a $dom(f)$. In addition, an interior point of a set is a point where it has an open neighborhood that is also in a set. But what is interior of a domain?
The domain of $f$ is a particular set, and the interior of the domain of $f$ is just the interior of that set. The rest of this answer explains what the interior of a set is, for completeness.
A picture from Wikipedia:
Let $S \subset \mathbb R^n$. Then the interior of $S$ is the set of all points $x \in S$ such that $B(x,\epsilon) \subset S$ for some $\epsilon > 0$. Here $B(x,\epsilon)$ is the ball of radius $\epsilon$ at $x$: $$ B(x,\epsilon) = \{ y \in \mathbb R^n \mid \| x - y \| < \epsilon \}. $$
So, in other words, $x$ is in the interior of $S$ if and only if there exists a number $\epsilon > 0$ such that all points within $\epsilon$ of $x$ also belong to $S$.