# 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)