  We study the continuous CM-regularity of torsion-free coherent sheaves on
polarized irregular smooth projective varieties $(X,\mathcal{O}_X(1))$, and its
relation with the theory of generic vanishing. This continuous variant of the
Castelnuovo-Mumford regularity was introduced by Mustopa, and he raised the
question whether a continuously $1$-regular such sheaf $\mathcal{F}$ is GV.
Here we answer the question in the affirmative for many pairs
$(X,\mathcal{O}_X(1))$ which includes the case of any polarized abelian
variety. Moreover, for these pairs, we show that if $\mathcal{F}$ is
continuously $k$-regular for some integer $1\leq k\leq \dim X$, then
$\mathcal{F}$ is a GV$_{-(k-1)}$ sheaf. Further, we extend the notion of
continuous CM-regularity to a real valued function on the $\mathbb{Q}$-twisted
bundles on polarized abelian varieties $(X,\mathcal{O}_X(1))$, and we show that
this function can be extended to a continuous function on
$N^1(X)_{\mathbb{R}}$. We also provide syzygetic consequences of our results
for $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ on $\mathbb{P}(\mathcal{E})$
associated to a $0$-regular bundle $\mathcal{E}$ on polarized abelian
varieties. In particular, we show that
$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ satisfies $N_p$ property if the
base-point freeness threshold of the class of $\mathcal{O}_X(1)$ in $N^1(X)$ is
less than $\frac{1}{p+2}$. This result is obtained using a theorem in the
Appendix written by Atsushi Ito.
