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

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.