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Intermediate value theorem for analytic functions on a Levi-Civita field. (English) Zbl 1181.26044

Summary: The proof of the intermediate value theorem for power series on a Levi-Civita field will be presented. After reviewing convergence criteria for power series [cf. the authors, in: \(p\)-adic functional analysis. Proceedings of the 6th international conference, Ioannina, Greece, 2000. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 222, 283–299 (2001; Zbl 0985.26014)], we review their analytical properties [cf. the authors, Ann. Math. Blaise Pascal 12, No. 2, 309–329 (2005; Zbl 1087.26020)]. Then we state and prove the intermediate value theorem for a large class of functions that are given locally by power series and contain all the continuations of real power series: using iteration, we construct a sequence that converges strongly to a point at which the intermediate value will be assumed.

MSC:

26E30 Non-Archimedean analysis
30G06 Non-Archimedean function theory
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
12J25 Non-Archimedean valued fields
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Full Text: Euclid