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Analyticity of almost everywhere differentiable functions. (English) Zbl 0705.30001

In 1931 A. S. Besicovitch proved the following two results for a complex- valued function f defined in a domain D of the complex plane:
1. If f is bounded in D and differentiable at all points of \(D\setminus E\), where E has zero one-dimensional Hausdorff measure, then f agrees in \(D\setminus E\) with a function which is analytic in D.
2. If f is continuous in D and differentiable at all points of \(D\setminus E\), where E has \(\sigma\)-finite one-dimensional Hausdorff measure, then f is analytic in D.
Note that here E is quite general, in particular it need not be closed. The author presents a new and simpler proof for these results. In fact, Besicovitch’s theorems are a rather quick consequence of a general result on interval functions in \({\mathbb{R}}^ n\). It tells that under certain conditions non-negativity of the lower derivative outside an exceptional set implies the non-negativity of the interval function itself. The proof of this employs partitioning and covering arguments with dyadic cubes. The methods are related and motivated by the work of W. F. Pfeffer in connection of generalized integrals in higher dimensions.
Reviewer: P.Mattila

MSC:

30B40 Analytic continuation of functions of one complex variable
28A10 Real- or complex-valued set functions
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