# Observability of continuity for a RCLL process

Let $$(\Omega, \mathcal{F}, P)$$ be a probability space and let $$X:[0;+\infty)\times\Omega\to\mathbb{R}^d$$ be a stochastic process on this probability space with the property that all its sample paths are RCLL. Let $$t_0>0$$. Let $$A=\{\omega\in\Omega | t\mapsto X_t(\omega) \text{ is a continuous function on } [0;t_0)\}$$. Is it possible to prove that $$A\in\mathcal{F}^X_{t_0}$$, where $$\left(\mathcal{F}^X_t\right)_{t\geq 0}$$ is the filtration generated by process $$X$$, i.e. $$\mathcal{F}^X_t=\sigma(X_s;0\leq s \leq t)$$ ?

This is a question I encountered in Karatzas & Shreve (Exercise 1.1.7.).

Without loss of generality, $$d=1$$; if not examine the coordinates of $$X$$ separately, or consider the rcll $$(\mathcal F^X_t)$$-adapted process $$t\mapsto|X_t|$$.
Consider the oscillation $$Y_t(\omega,\delta):=\sup_{s:|t-s|\le\delta}X_s(\omega)-\inf_{s:|t-s|\le\delta}X_s(\omega)$$. Because $$X$$ is rcll, the supremum and infimum here can be taken over rational $$s$$. Consequently, $$Y_t(\omega,\delta)$$ is $$\mathcal F_{t+\delta}^X$$-measurable.
Fix $$t>0$$ and define \eqalign{ B(t):&=\{\omega: X_{s-}(\omega)\not=X_s(\omega)\hbox{ for some }s\in[0,t]\}\cr &=\cup_{m=1}^\infty\cap_{n=1}^\infty\cup_{k=1}^n\{\omega: Y_{t(k-1)/n}(\omega,1/n)>1/m\}.\cr } (In effect, if $$X$$ has a jump in the time interval $$[0,t]$$ of size exceeding $$1/m$$, then for each $$n$$ the oscillation of $$X$$ over $$[t(k-1)/n,t/n]$$ must exceed $$1/m$$ for some $$k\in\{1,2,\ldots,n\}$$.) Notice that $$B(t)$$ is $$\mathcal F^X_{t+}$$-measurable. Finally, $$A=\cup_{q