Hartogs' theorem

Note that the terminology is inconsistent and Hartogs' theorem may also mean Hartogs' lemma on removable singularities, the result on Hartogs number in axiomatic set theory, or Hartogs extension theorem.

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 $F:{\textbf{C}}^n \to {\textbf{C}}$ is a function which is analytic in each variable z_i, 1 ≤ i ≤ n, while the other variables are held constant, then F is 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 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 $f \colon {\textbf{R}}^n \to {\textbf{R}}$ 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.)

References

Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

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