Let $X$ be a stochastic process (parametrised by positive reals) whose sample paths are RCLL (finite left hand limits everywhere, right continuous everywhere) almost surely. Let $\{\mathcal{F}_t:t\geq 0\}$ be a filtration satisfying $\mathcal{F}_t^X\subset \mathcal{F}_t$, $t\geq 0$, and also $\mathcal{F}_{t_0}$ is complete under $P$. Show that the event that $X$ is continuous on $[0,t_0)$ measurable with respect to $\mathcal{F}_{t_0}$.
I understand that an RCLL function is continuous if and only if its restriction to a countable dense set is uniformly continuous, and if we fix the countable dense set to be the rationals in $[0,t_0)$, then we can show that the event of the restriction being uniformly continuous is measurable with respect to $\mathcal{F}_{t_0}$. If $N\in \mathcal{F}$ denotes the event that the function is not RCLL, then $N$ may not be measurable with respect to $\mathcal{F}_{t_0}$, even though $\mathcal{F}_{t_0}$ is complete because completeness only says that every subset of a $P$-null measurable set in $\mathcal{F}_{t_0}$ is also measurable with respect to the same $\sigma$-algebra. The solution given in the book just writes that $\{\text{function is continuous}\}=\{\text{Restriction to rationals is uniformly continuous}\}\cap N^c$, which somehow seems to assume that $N$ is measurable with respect to $\mathcal{F}_{t_0}$.
