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Tuesday, December 3, 2013

**Abstract:** Functions definable in o-minimal structures behave rather nicely except for a "small" set of the ambient space they are defined. In particular for any function $f$ on $R$, the universe of an o-minimal structure, and almost every nice property of functions and every point $a \in R$ there is an open interval with $a$ in its frontier, such that $f$ has that property inside the interval. It is to be noted that quasianalitycity of a system of functions does not imply directly o-minimality of the structure they generate. We then consider a set of conditions on a quasianalytic system of algebras of functions together with an $\mathbb{R}$-algebra embedding from their germs to the ring of generalized power series. If these germs together with the embedding satisfy certain properties, we get o-minimality of the structure the original functions generate.