On extendibility of a map induced by Bers isomorphism

Let $S$ be a closed Riemann surface of genus $g(\geqq 2)$ and set $\dot{S}=S \setminus {\hat{z}_0 }$\mathrm{.}Thenwehavethecomposedmap$\varphi\circ r$ of a map $r:T(S)\times U\to F(S)$ and the Bers isomorphism $\varphi: F(S) \rightarrow T(\dot{S})$,where$F(S)$istheBersfiberspaceof$S$,$T(X)$ is the Teichm"uller space of $X$ and $U$ is the upper half-plane. The purpose of this paper is to show the map $\varphi\circ r:T(S)\times U \rightarrow T(\dot{S})$\mathrm{.}hasacontinuousextensiontosomesubsetoftheboundary$T(S) \times \partial U$.