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