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

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

Automated summary

The two assertions, one about a definable bijection between bounded and unbounded intervals, and the other about the existence of definable continuous extensions for functions defined on closed subsets, are equivalent for an o-minimal expansion of an ordered group M.