An access theorem for analytic functions

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 $t \in [d(K),1)$.