Submission date: 6 October 2020

Abstract:

We consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group (DCULOAS structure) in this paper. The first main theorem is the following monotonicity theorem. For a definable function f on an interval I, the interval I is decomposed into four definable sets. Three sets are open definable sets on which f is locally constant, locally strictly increasing and continuous, and locally strictly decreasing and continuous, respectively. The last definable set is discrete and closed.
We also investigate uniform continuous definable functions and derive Arzela-Ascoli-type theorem for definable functions. Consider the parameterized function f:C × P → M which is equi-continuous with respect to P. The projection image of the set at which f is discontinuous to the parameter space P is of dimension smaller than dim P when C is closed and bounded.
Finally, we demonstrate that an archimedean DCULOAS structure which enjoys definable Tietze extension property is o-minimal.

Mathematics Subject Classification: 03C64

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Full text arXiv 2010.02420: pdf, ps.