Hartogs's theorem on separate holomorphicity
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 formula_1 is a function which is analytic in each variable "z", 1 ≤ "i" ≤ "n", while the other variables are held constant, then "F" is a continuous function. 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. If we assume that a function formula_2 is differentiable (or even analytic) in each variable separately, it is not true that formula_3 will necessarily be continuous. A counterexample in two dimensions is given by If in addition we define formula_5, this function has well-defined partial derivatives in formula_6 and formula_7 at  the origin, but it is not continuous at origin. (Indeed, the limits along the lines formula_8 and formula_9 are not equal, so there is no way to extend the definition of formula_3 to include the origin and have the function be continuous there.)