Continuous Non-extendability and Holomorphic Extendability of Proper Holomorphic Mappings

We study the extendability problems for proper holomorphic mappings between, possibly different dimensional, domains. After reviewing the known results, we show the following: There exists a proper holomorphic mapping between generalized pseudoellipsoids with different dimensions which can not be extended continuously across the boundary. If we assume the holomorphic extendability, the mappings have normal forms up to automorphisms of the source and the target domains.