This is a follow up to this question of mine, this follow up question appears to be asked even in this other question but, to my knowledge, without an answer.
Here is the question:
Let $X$ be a stochastic process whose sample paths are RCLL a.s. Let $A$ be the event that $X$ is continuous on $[0,t_0)$. Show that $A$ can fail to be in $\mathcal{F}_{t_0}^X$. But if $\{\mathcal{F_t;t\ge 0}\}$ is a filtration satisfying $\mathcal{F}^X_t\subset\mathcal{F}_t,t\ge 0$ and $\mathcal{F}_{t_0}$ is complete under P, then $A \in \mathcal{F}_{t_0}$.
Where $\mathcal{F}_{t_0}^X:=\sigma{(\{X_s;0\le s\le t_0\})}$.
My naive reasoning:
If the sample paths are RCLL a.s. this means we can take a $t_1 \in [0,t_0)$ s.t. at $t_1$ the sample paths $X(w)_{t_1}$ have a jump discontinuity for any $w \in \Omega$. So it seems that $A$ can easily fail to be in $\mathcal{F}_{t_0}^X$.
But this must be wrong because taking a complete filtration does not fix the problem with my line of reasoning.
How is the question really solved?
