# Hartogs's 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 $F:{\textbf {C}}^{n}\to {\textbf {C}}$ is a function which is analytic in each variable zi, 1 ≤ in, while the other variables are held constant, then F is a continuous function.
Note that there is no analogue of this theorem for real variables. If we assume that a function $f\colon {\textbf {R}}^{n}\to {\textbf {R}}$ is differentiable (or even analytic) in each variable separately, it is not true that $f$ will necessarily be continuous. A counterexample in two dimensions is given by
$f(x,y)={\frac {xy}{x^{2}+y^{2}}}.$ If in addition we define $f(0,0)=0$ , this function has well-defined partial derivatives in $x$ and $y$ at the origin, but it is not continuous at origin. (Indeed, the limits along the lines $x=y$ and $x=-y$ are not equal, so there is no way to extend the definition of $f$ to include the origin and have the function be continuous there.)