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