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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.

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 |

### Keywords:

Hausdorff measure
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\textit{E. J. Howard}, Proc. Am. Math. Soc. 110, No. 3, 745--753 (1990; Zbl 0705.30001)

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### References:

[1] | A. S. Besicovitch, On sufficient conditions for a function to be analytic, and behaviour of analytic functions in the neighborhood of non-isolated singular points, Proc. London Math. Soc. 32 (1931), 1-9. |

[2] | K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. · Zbl 0587.28004 |

[3] | Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801 |

[4] | Ralph Henstock, Lectures on the theory of integration, Series in Real Analysis, vol. 1, World Scientific Publishing Co., Singapore, 1988. · Zbl 0668.28001 |

[5] | W. F. Pfeffer, A note on the lower derivate of a set function and semihereditary systems of sets, Proc. Amer. Math. Soc. 18 (1967), 1020 – 1025. · Zbl 0153.38406 |

[6] | W. F. Pfeffer, On the lower derivate of a set function, Canad. J. Math. 20 (1968), 1489 – 1498. · Zbl 0169.38501 |

[7] | Washek F. Pfeffer, The multidimensional fundamental theorem of calculus, J. Austral. Math. Soc. Ser. A 43 (1987), no. 2, 143 – 170. · Zbl 0638.26011 |

[8] | -, Divergence theorem for vector fields with singularities, unpublished. · Zbl 0719.26009 |

[9] | Washek F. Pfeffer, Stokes theorem for forms with singularities, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 14, 589 – 592 (English, with French summary). · Zbl 0663.26009 |

[10] | W. F. Pfeffer and W. J. Wilbur, A note on cluster points of a semihereditary stable system of sets., Proc. Amer. Math. Soc. 21 (1969), 121 – 125. · Zbl 0183.31802 |

[11] | Washek F. Pfeffer and Wei-Chi Yang, A multidimensional variational integral and its extensions, Real Anal. Exchange 15 (1989/90), no. 1, 111 – 169. · Zbl 0704.26017 |

[12] | H. L. Royden, A generalization of Morera’s theorem, Ann. Polon. Math. 12 (1962), 199 – 202. · Zbl 0105.28103 |

[13] | Stanisław Saks, Theory of the integral, Second revised edition. English translation by L. C. Young. With two additional notes by Stefan Banach, Dover Publications, Inc., New York, 1964. · Zbl 1196.28001 |

[14] | Lawrence Zalcman, Analyticity and the Pompeiu problem, Arch. Rational Mech. Anal. 47 (1972), 237 – 254. · Zbl 0251.30047 |

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