In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter.
Continuity is a nice property for (the sample paths of) a process to have, since it implies that they are well-behaved in some sense, and, therefore, much easier to analyze.
It is implicit here that the index of the stochastic process is a continuous variable.
Some authors[1] define a "continuous (stochastic) process" as only requiring that the index variable be continuous, without continuity of sample paths: in another terminology, this would be a continuous-time stochastic process, in parallel to a "discrete-time process".
Even worse, even if A is an event, P(A) can be strictly positive even if P(At) = 0 for every t ∈ T. This is the case, for example, with the telegraph process.