Let $X$ be a predictable process defined on some filtered probability space (as good as possible) such that
$$X_t \in \{0,1\},\quad \forall t\ge 0.$$
Does this imply the left continuity of $X$? If so, is there a reference or a proof?
In the following example, all random variables are constant with respect to $\omega$, so $\Omega$ could be taken to be a point mass.
For each $n$, let $$X_t^n = \begin{cases} 0, & t \le 1-1/n \\ 1, & t > 1-1/n.\end{cases}$$ Then $X_t^n$ is adapted (trivially) and left continuous, so it is predictable. Let $X_t = \lim_{n \to \infty} X_t^n$ which is thus also predictable. But $$X_t = \begin{cases} 0, & t < 1 \\ 1, & t \ge 1\end{cases}$$ which is not left continuous.