## Property of a Markov Chain

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.