# Functions nice in each variable separately

Consider the following question. If ${f:{\mathbb R}^2\rightarrow {\mathbb R}}$ is continuous in each variable separately, need it be continuous in both? Although Cauchy (incorrectly) said it should, note it is not even obvious that it should be measurable.

A classical example is

$\displaystyle f(x,y):= \frac {xy}{x^2+y^2}$

Defining ${f(0,0)=0}$, this function is real-analytic in each variable separately, but it is not even continuous at the origin.

By contrast, in the complex-analytic situation we have the following result.

Theorem 1 (Hartogs) Suppose ${f:{\mathbb C}^{n_1}\times {\mathbb C}^{n_2}\rightarrow {\mathbb C}}$ is holomorphic in each variable separately. Then ${f}$ is holomorphic on ${{\mathbb C}^{n_1+n_2}}$.

Below the fold, I will state some more results of this type and prove some of them. Continue reading