Before revision:   A weaker sufficient condition than that of \mbox{%DIFAUXCMD \cite[Section V.2]{FlemingS for the continuity of the value function is obtained.
After revision:  The goal is to find an optimal control that minimizes the total cost up to a stochastic exit time. A weaker sufficient condition than that of \mbox{%DIFAUXCMD \cite[Section V.2]{FlemingS for the continuity of the value function is obtained.