This is part of a proof of a Theorem from Schilling's Brownian Motion.
Let $A_s$ be a continuous weakly increasing process. Given a partition of points $t_j$, define random times $\tau_j(\omega):=\inf\{s\ge 0: A_s (\omega)>jT/n\}\wedge T,j\ge 1,$ and $\tau_0(\omega)=0$.
In this case, how does the continuity of $s\mapsto A_s$ ensure that $A_{\tau_j}-A_{\tau_{j-1}}=T/n$ for all $1\le j \le \lfloor nA_T/T\rfloor$?
This identity clearly holds in the case that $\tau_j(\omega) <T$, but how do we ensure this when we might have $\tau_j$ or $\tau_{j-1}=T$?