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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 counterexample to access theorems for $ C\sp \infty$ functions


Author: Marvin Ortel
Journal: Proc. Amer. Math. Soc. 123 (1995), 819-825
MSC: Primary 26B99; Secondary 26E10, 30G12, 31A20, 57R45
MathSciNet review: 1239802
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Abstract: We construct a function $ f:{\mathbb{R}^2} \to \mathbb{R}$ with the following properties:

(1) f is of class $ \infty $.

(2) If $ m \in {\mathbb{R}^2}$, then $ \gamma :[0,1] \to {\mathbb{R}^2}$ exists such that $ \gamma (0) = m,\gamma $ is continuous, and $ f \circ \gamma $ is strictly increasing on [0, 1].

(3) If $ \sigma :[0,1) \to {\mathbb{R}^2}$ is continuous and $ f \circ \sigma $ is nondecreasing on [0, 1), then $ \sup \{ \vert\sigma (s)\vert:0 \leq s < 1\} < \infty $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1239802-9
PII: S 0002-9939(1995)1239802-9
Keywords: Potential theory, boundary behavior, singularities of differentiable mappings, real-analytic functions
Article copyright: © Copyright 1995 American Mathematical Society