Characterization of continuity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds

For a compact subset in a compact Hermitian manifold, we prove that the continuity of the extremal function at a given point in the set is a local property and that the continuity of a weighted extremal function follows from the continuities of the extremal function and the weight function. These results are generalizations of the results of Nguyen \cite{Ng24} on compact Kähler manifolds. Moreover, for a compact subset in a compact Hermitian manifold, at the point level and accordingly at the global level, we characterize the continuity of the extremal function via the local $L$-regularity, which is equivalent to the weak local $L$-regularity. We also show that the $L$-regularity of a compact subset in ${\mathbb{C}}^n$ at a star center implies the local $L$-regularity. Consequently, a convex compact $L$-regular subset in ${\mathbb{C}}^n$ is locally $L$-regular.