Continuity theorem

continuity principle

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c0255901.png" /> be a domain of holomorphy in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c0255902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c0255903.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c0255904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c0255905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c0255906.png" /> be two sequences of sets, with compact closures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c0255907.png" />, for which the maximum modulus principle holds for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c0255908.png" /> that are holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c0255909.png" />, that is,

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559010.png" />

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559011.png" />

Then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559012.png" /> converges to some bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559014.png" /> to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559015.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559017.png" /> has compact closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559019.png" /> has compact closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559020.png" />. If for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559021.png" /> one takes analytic hypersurfaces and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559022.png" /> their boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559023.png" />, one obtains the Behnke–Sommer theorem (see [1]). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559024.png" /> a Jordan curve of the form

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559025.png" />

is given. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559027.png" />, be a family of domains in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559028.png" />-plane having the property that for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559029.png" /> there is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559031.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559032.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559033.png" /> is holomorphic at the points of the "discs"

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559034.png" />

and at one point of the limiting "disc"

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559035.png" />

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559036.png" /> 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.

References

[1]	H. Behnke, P. Thullen, "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer (1970) (Elraged & Revised Edition. Original: 1934)
[2]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[3]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian) MR Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL

Comments

The continuity principle is also known as Hartogs' Kontinuitätssatz (Hartogs' continuity theorem).

References

[a1]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 3 MR0635928 Template:ZBL
[a2]	R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 2 MR0847923 Template:ZBL