  We prove that the regularity of the extremal function of a compact subset of
a compact K\"ahler manifold is a local property, and that the continuity and
H\"older continuity are equivalent to classical notions of the local
$L$-regularity and the locally H\"older continuous property in pluripolential
theory. As a consequence we give an effective characterization of the
$(\Cc^\al, \Cc^{\al'})$-regularity of compact sets, the notion introduced by
Dinh, Ma and Nguyen. Using this criterion all compact fat subanalytic subsets
in $\bR^n$ are shown to be regular in this sense.
