A counterexample to access theorems for $C^ \infty$ functions
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- by Marvin Ortel
- Proc. Amer. Math. Soc. 123 (1995), 819-825
- DOI: https://doi.org/10.1090/S0002-9939-1995-1239802-9
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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 $\sup \{ |\sigma (s)|:0 \leq s < 1\} < \infty$.References
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Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 819-825
- MSC: Primary 26B99; Secondary 26E10, 30G12, 31A20, 57R45
- DOI: https://doi.org/10.1090/S0002-9939-1995-1239802-9
- MathSciNet review: 1239802