Hartogs's theorem on separate holomorphicity: Difference between revisions

Revision as of 16:48, 15 April 2022

Revision as of 16:19, 17 April 2022

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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]].

There is no analogue of this [[theorem]] for [[Function of several real numbervariables|real]] variables]]. If we assume that a function

<math>f \colon {\textbf{R}}^n \to {\textbf{R}}</math>

is [[Differentiable function|differentiable]] (or even [[analytic function|analytic]]) in each variable separately, it is not true that <math>f</math> will necessarily be continuous. A counterexample in two dimensions is given by