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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32A40

Retrieve articles in all journals with MSC: 32A40

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society