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In [[mathematics]], '''Hartogs's theorem''' is a fundamental result of [[Friedrich Hartogs]] in the theory of [[several complex variables]]. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if <math>F:{\textbf{C}}^n \to {\textbf{C}}</math> is a function which is [[analytic function|analytic]] in each variable ''z''<sub>''i''</sub>, 1 ≤ ''i'' ≤ ''n'', while the other variables are held constant, then ''F'' is a [[continuous function]].
A [[corollary]] is that the function ''F'' is then in fact an analytic function in the ''n''-variable sense (i.e. that locally it has a [[Taylor expansion]]). Therefore, 'separate analyticity' and 'analyticity' are coincident notions, in the theory of several complex variables.
Starting with the extra hypothesis that the function is continuous (or bounded), the theorem is much easier to prove and in this form is known as [[Osgood's lemma]].
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* http://planetmath.org/hartogsstheoremonseparateanalyticity
{{PlanetMath attribution|id=6024|title=Hartogs's theorem on separate analyticity}}
[[Category:Several complex variables]]
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