Consider the following question. If 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

Defining , 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 is holomorphic in each variable separately. Then is holomorphic on .

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