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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An access theorem for analytic functions


Author: Marvin Ortel
Journal: Trans. Amer. Math. Soc. 347 (1995), 2213-2223
MSC: Primary 32C05; Secondary 31B05
DOI: https://doi.org/10.1090/S0002-9947-1995-1273513-3
MathSciNet review: 1273513
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Abstract: Suppose that $\mathcal {M}$ is an analytic manifold, ${m_0} \in \mathcal {M},f:\mathcal {M} \to \mathbb {R}$, and $f$ is analytic. Then at least one of the following three statements is true: (1) ${m_0}$ is a local maximum of $f$. (2) There is a continuous path $\sigma :[0,1] \to \mathcal {M}$ such that $\sigma (0) = {m_0}$, $f \circ \sigma$ is strictly increasing on $[0,1]$, and $\sigma (1)$ is a local maximum of $f$. (3) There is a continuous path $\sigma :[0,1) \to \mathcal {M}$ with these properties: $\sigma (0) = {m_0};f \circ \sigma$ is strictly increasing on $[0,1)$; whenever $K$ is a compact subset of $\mathcal {M}$, there is a corresponding number $d(K) \in [0,1)$ such that $\sigma (t) \notin K$ for all $t \in [d(K),1)$.


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Keywords: Real-analytic functions, analytic manifolds, singularities
Article copyright: © Copyright 1995 American Mathematical Society