View source for 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|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,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559010.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559011.png" /></td> </tr></table>

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 [[#References|[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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559025.png" /></td> </tr></table>

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" 

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and at one point of the limiting  "disc" 

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025590/c02559035.png" /></td> </tr></table>

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|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|Hartogs theorem]] on removable singularities (1916) for holomorphic functions of several complex variables.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Behnke,   P. Thullen,   "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer  (1970)  (Elraged &amp; Revised Edition. Original: 1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR></table>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.G. Krantz,   "Function theory of several complex variables" , Wiley  (1982)  pp. Chapt. 3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Range,   "Holomorphic functions and integral representation in several complex variables" , Springer  (1986)  pp. Chapt. 2</TD></TR></table>