Given a continuous time homogeneous Markov chain with a standard semi-group of transition probabilities (e.g. all the terms of the matrix are continuous functions of ). How do you prove that for any pair of states:

Either we have for all or for all ?

While we can still use the continuity when dealing with a single matrix element, we can't use directly the semi-group property to propagate any property to the whole time interval.

Thanks for your help in advance.