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{{dablink|Note that the terminology is inconsistent and Hartogs' theorem may also mean [[Hartogs' lemma]] on removable singularities, the result on [[Hartogs number]] in axiomatic set theory, or [[Hartogs extension theorem]].}}

In [[mathematics]], '''Hartogs' 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 an [[analytic function]] 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]] of this is that ''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 several complex variables theory.

The theorem with the extra condition that the function is continuous (or bounded) is much easier to prove and is known as [[Osgood's lemma]].

Note that there is no analogue of this [[theorem]] for [[real number|real]] variables. If we assume that a function
<math>f \colon {\textbf{R}}^n \to {\textbf{R}}</math>
is [[differentiable]] (or even [[analytic function|analytic]]) in each variable separately, it is not true that <math>f</math> will necessarily be continuous. A counterexample in two dimensions is given by

:<math>f(x,y) = \frac{xy}{x^2+y^2}.</math>

This function has well-defined [[partial derivative]]s in <math>x</math> and <math>y</math> at 0, but it is not [[Continuous function|continuous]] at 0 (the [[limit of a function|limits]] along the lines <math>x=y</math> and <math>x=-y</math> give different results).

== References ==
* Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992.

{{PlanetMath attribution|id=6024|title=Hartogs's theorem on separate analyticity}}

==External links==
* [http://www.encyclopediaofmath.org/index.php/Hartogs_theorem Hartogs theorem] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

[[Category:Several complex variables]]
[[Category:Theorems in complex analysis]]

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