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.
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.
There is no analogue of this theorem for real variables.
is differentiable (or even analytic) in each variable separately, it is not true that
This article incorporates material from Hartogs's theorem on separate analyticity on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.