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 $\tau$?



Continuity w.r.t time vs Continuity w.r.t. stopping times
A process which is right continuous can still be left continuous *at a fixed time* almost surely. For example, a Poisson process $N=(N\_t)\_{t \ge 0}$ is right continuous, and for any (deterministic) time $t$ it holds that $\lim\_{s \uparrow t}N\_s=N\_t$ a.s., simply because $\mathbb{P}(N \text{ jumps at time } t)=0$. It is in general a stronger statement to say the same is true for stopping times, and this property is known as *quasi-left-continuity*. As usual, George Lowther's blog has a [nice post about this](https://almostsure.wordpress.com/2010/07/19/properties-of-feller-processes/), including the proof that Feller processes are quasi-left-continuous.


