Continuous CM-regularity of semihomogeneous vector bundles

We show that if $X$ is an abelian variety of dimension $g\ge 1$ 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.