Let $f:\mathbb{R}^n\to \mathbb{R}$ be a convex function. It is well-known that (see e.g.~here) that there exists a set $A$ such that $$ f(x)=\sup_{a\in A}A(a)x+b(a), \quad \forall x\in \mathbb{R}^n, $$ where $A(a)\in\mathbb{R}^{n\times n}$ and $b(a)\in \mathbb{R}^n$.
I was wondering, is there any reference saying we can choose $A$ as a countable set?
This is easy. $\{(A(a),b(a)): a \in A\}$ is a subset of a separable metric space, so it is itself separable. Take a countable dense subset of this and use the fact that supremum of any set of real numbers is same as the supremum over countable dense subset.