ON SHIMODA?S THEOREM

The present work is devoted to Shimoda's Theorem on the holomorphicity of a function f(z, w) which is holomorphic by w is an element of V for each fixed z is an element of U and is holomorphic by z is an element of U for each fixed w is an element of E, where E subset of V is a countable set with at least one limit point in V. Shimoda proves that in this case f(z, w) is holomorphic in U xV except for a nowhere dense closed subset of U xV. We prove the converse of this result, that is for an arbitrary given nowhere dense closed subset of U, S subset of U, there exists a holomorphic function, satisfying Shimoda's Theorem on U xV subset of C-2, that is not holomorphic on S xV. Moreover, we observe conditions which imply empty exception sets on Shimoda's Theorem and prove generalizations of Shimoda's Theorem.