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
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.
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.
Note that there is no analogue of this theorem for real variables. If we assume that a function
If in addition we define