  For the open unit disc $\mathbb{D}$ in the complex plane, it is well known
that if $\phi \in C(\overline{\mathbb{D}})$ then its Berezin transform
$\widetilde{\phi}$ also belongs to $C(\overline{\mathbb{D}})$. We say that
$\mathbb{D}$ is BC-regular. In this paper we study BC-regularity of some
pseudoconvex domains in $\mathbb{C}^n$ and show that the boundary geometry
plays an important role. We also establish a relationship between the essential
norm of an operator in a natural Toeplitz subalgebra and its Berezin transform.
