Let $T\subseteq \overline{\mathbb R}$ be an intervall and $\tau:\Omega\to \overline T$ a stopping time. (I don't think it is necessary to mention that $\tau$ is a stopping time but I am not sure).
For fixed $\omega \in \Omega$, do we have that the mapping $$t\mapsto\tau(\omega)\land t$$ is continuous? $(t \in T)$
Can there arise any problems if $\tau(\omega)\in \overline T\setminus T?$