Michael's selection theorem for a mapping definable in an O-minimal structure defined on a set of dimesion 1

Abstract

Let $R$ be a real closed field and let some o-minimal structure extending $R$ be given. Let $F\colon X \rightrightarrows R^m$ be a definable multivalued lower semicontinuous mapping with nonempty definably connected values defined on a definable subset $X$ of $R^n$ of dimension $1$ ($X$ can be identified with a finite graph immersed in $R^n$). Then $F$ admits a definable continuous selection.