# Continuity theorem

continuity principle

Let be a domain of holomorphy in , , and let and , be two sequences of sets, with compact closures in , for which the maximum modulus principle holds for functions that are holomorphic in , that is,  Then if converges to some bounded set and to a set , and if and has compact closure in , then has compact closure in . If for one takes analytic hypersurfaces and for their boundaries , one obtains the Behnke–Sommer theorem (see ). Hence it follows that every domain of holomorphy is pseudo-convex. Applied to a specific function, certain modifications of the continuity theorem are known as theorems on "analytic discs" . For example, the strong theorem on analytic "discs" asserts the following. Suppose that in a Jordan curve of the form is given. Let , , be a family of domains in the -plane having the property that for any compact set there is a number such that for all . If is holomorphic at the points of the "discs" and at one point of the limiting "disc" then is holomorphic also at all points of the limiting "disc" . Theorems on "analytic discs" are very useful in the holomorphic extension of domains and in constructing envelopes of holomorphy (cf. Holomorphic envelope), for example, in the proof of Bochner's theorem on the envelope of holomorphy of a tube domain, of the Osgood–Brown theorem, and of the theorem on "imbedded edges" , "the edge-of-the-wedge" , "C-convex hulls" , and others. The continuity principles given go back to the Hartogs theorem on removable singularities (1916) for holomorphic functions of several complex variables.

How to Cite This Entry:
Continuity theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuity_theorem&oldid=24404
This article was adapted from an original article by V.S. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article