Definable Tietze extension property in o-minimal expansions of ordered groups

Abstract

The following two assertions are equivalent for an o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots)$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function $f:A \rightarrow M$ defined on a definable closed subset of $M^n$ has a definable continuous extension $F:M^n \rightarrow M$.