Random processes

Question

I hope someone could tell me how to explain that "random process is continuous by probability" and "random process is differentiated by probability"? I know that definitions are these:

1. Given a time $t ∈ T$, $X$ is said to be continuous in probability at $t_0$ if, for all $ε > 0$, $\lim_{h \to 0} P(∣X_{t_0+h}−X_{t_0}∣ >ε)=0$. If the process is continuous in probability at every point, then it is continuous in probability.

2. Process $X_t$ is differentiated by probability at $t_0$ if $\exists$ limit

   $\lim_{h \to 0} (X_{t_0 + h} - X_{t_0})/h $ by probability.

Thanks a lot!

I need to explain what "random process is continuous by probability" and "random process is differentiated by probability" mean.

Answer 1

1 means that for every $t_0$, the probability of a discontinuity happening at $t_0$ is zero. That doesn't mean, by the way, that there are no discontinuities anywhere; it only says that there are no "scheduled" discontinuities.