Several places in "Optimal Stopping and Free-Boundary Problems" Peskir and Shiryaev make the assumption that a (Markov) process $X = (X_t)_{t\geq 0}$ has sample paths which are right continuous and left continuous over stopping times.

What is the reason for making the distinction between continuity with respect to the time $t$ versus continuity with respect to a stopping time $\tau$? If a process is right and left continuous with respect to $t$ isn't it automatically right and left continuous with respect to a stopping time $\tau$?