(Cinlar, probability and stochastics, 5.7.29): Let $T_a = \inf[t > 0: W_t \ge a]$, and $S_a = \inf[t > 0: W_t > a]$, where $W_t$ is Wiener. Show that $a \to T_a$ is left-continuous and $a \to S_a$ is right-continuous, and that $T_a = S_{a-}$.
I am not familiar with showing the continuity of the $\inf$ set. Can you gives some hint how to proceed? I am not sure if it helps, but the previous exercise is about showing $T_a = S_a$ a.s..