10 research outputs found
Monotone functions and maps
In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2
(2011)] we defined semi-monotone sets, as open bounded sets, definable in an
o-minimal structure over the reals, and having connected intersections with all
translated coordinate cones in R^n. 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 Basu et al. for semi-monotone
sets.Comment: 30 pages. Version 2 appeared in RACSAM. In version 3 Corollaries 1
and 2 were corrected. In version 4 Theorem 3 is correcte