In  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  for semi-monotone sets.
|Number of pages||29|
|Journal||Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas|
|Early online date||8 Jul 2012|
|Publication status||Published - 2013|
Basu, S., Gabrielov, A., & Vorobjov, N. (2013). Monotone functions and maps. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 107(1), 5-33. https://doi.org/10.1007/s13398-012-0076-4