Hartogs's theorem on separate holomorphicity
NB that the terminology is inconsistent and Hartogs' theorem may also mean Hartogs' lemma on removable singularities, or the result on Hartogs number
In mathematics, Hartogs' theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. It states that for complex-valued functions F on Cn, with n > 1, being an analytic function in each variable zi, 1 ≤ i ≤ n, while the others are held constant, is enough to prove F 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.