Let $X$ be a Hausdorff separated locally convex space with topological dual $X^*$ and let $C\subset X$ be non-empty closed convex. The support functional of $C$ is defined by
$$ s_C:X^*\to\mathbb{R}\cup\{+\infty\},\ s_C(x^*)=\sup_{x\in C}x^*(x) $$
I know that $s_C$ is weak-star continuous at least at one point iff $C$ is finite-dimensional and free of lines.
I wonder if a characterization exists for the continuity of $s_C$ at least at one point with respect to the Mackey topology on $X^*$ relative to the duality $(X^*,X)$.