  In this paper we prove a general result of the ``Hopf lemma'' type for CR
mappings, with nonidentically vanishing Jacobians, between real hypersurfaces
in C^n with smooth or real analytic boundaries. Applications of this result to
finiteness and holomorphic extendibility of such mappings are also given. The
novelty here is that we make no assumption on the nonflatness of the mapping or
its Jacobian, nor do we assume that the hypersurfaces are pseudoconvex or
minimally convex.
