AN ACCESS THEOREM FOR ANALYTIC-FUNCTIONS

Suppose that M is an analytic manifold, m(0) is an element of M, f : M --> 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] --> M such that sigma(0) = mo, f circle 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) --> M with these properties: sigma(0) = m(0); f circle sigma is strictly increasing on [0, 1); whenever K is a compact subset of M, there is a corresponding number d(K) is an element of [0, 1) such that sigma(t) is not an element of K for all t is an element of [d(K), 1).