Boundary regularity of correspondences in pmb{{mbf C}^n}

Let $M, M'$ be smooth, real analytic hypersurfaces of finite type in ${\mbf C^n}$ and $\h f$ a holomorphic correspondence (not necessarily proper) that is defined on one side of $M$, extends continuously up to $M$ and maps $M$ to $M'$. It is shown that $\h f$ must extend across $M$ as a locally proper holomorphic correspondence. This is a version for correspondences of the Diederich--Pinchuk extension result for CR maps.