Title: Continuous CM-regularity of semihomogeneous vector bundles,
Abstract: We show that if $X$ is an abelian variety of dimension $g \geq 1$ and
${\mathcal E}$ is an M-regular coherent sheaf on $X$, the Castelnuovo-Mumford
regularity of ${\mathcal E}$ with respect to an ample and globally generated
line bundle ${\mathcal O}(1)$ on $X$ is at most $g$, and that equality is
obtained when ${\mathcal E}^{\vee}(1)$ is continuously globally generated. As
an application, we give a numerical characterization of ample semihomogeneous
vector bundles for which this bound is attained.