Discontinuity set of the expected value of a continuous process

Let $X_t$ be a continuous real valued stochastic process on $mathbb R_+$. Then it is not necessarily true that $E(X_t)$ is continuous in $t$.

Question:

What is known about the discontinuity set of $E(X_t)$? Namely:

  • Can it have removable discontinuities?

  • Can it have essential discontinuities?

  • Can it be discontinuous everywhere?

  • What is the analytic class of the discontinuity set, if if is known? (eg, $G_{sigma delta}$, $F_delta$, etc)