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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A holomorphic function with wild boundary behavior


Author: Josip Globevnik
Journal: Proc. Amer. Math. Soc. 93 (1985), 648-652
MSC: Primary 32A40
MathSciNet review: 776196
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Abstract: Let $ B$ be the open unit ball in $ {{\mathbf{C}}^N},N > 1$. It is known that if $ f$ is a function holomorphic in $ B$, then there are $ x \in \partial B$ and an arc $ \Lambda $ in $ B \cup \left\{ x \right\}$, with $ x$ as one endpoint along which $ f$ is constant. We prove

Theorem. There exist an $ r > 0$ and a function $ f$ holomorphic in $ B$ with the property that, if $ x \in \partial B$ and $ \Lambda $ is a path with $ x$ as one endpoint, such that $ \Lambda - \left\{ x \right\}$ is contained in the open ball of radius $ r$ which is contained in $ B$ and tangent to $ \partial B$ at $ x$, then $ \lim_{z \in \Lambda ,z \to x}f\left( z \right)$ does not exist.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0776196-0
PII: S 0002-9939(1985)0776196-0
Article copyright: © Copyright 1985 American Mathematical Society