Monotone functions and maps

Saugata Basu, Andrei Gabrielov, Nicolai Vorobjov

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

In [1] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals (e.g., real semialgebraic or subanalytic sets), and having connected intersections with all translated coordinate cones in Rn . In this paper we develop this theory further by defining monotone functions and maps, and studying their fundamental geometric properties. We prove several equivalent conditions for a bounded continuous definable function or map to be monotone. We show that the class of graphs of monotone maps is closed under intersections with affine coordinate subspaces and projections to coordinate subspaces. We prove that the graph of a monotone map is a topologically regular cell. These results generalize and expand the corresponding results obtained in [1] for semi-monotone sets.
Original languageEnglish
Pages (from-to)5-33
Number of pages29
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
Volume107
Issue number1
Early online date8 Jul 2012
DOIs
Publication statusPublished - 2013

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