Given a continuous time homogeneous Markov chain with a standard semi-group of transition probabilities (P(t))_{t\geq 0} (e.g. all the terms of the matrix P(t) are continuous functions of t). How do you prove that for any pair (i,j) of states:
Either we have P_{ij}(t)=0 for all t>0 or P_{ij}(t)>0 for all t>0?

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.