Let $\mathbb{S}$ be countable and $\Omega$ be the set of all right continuous functions $\omega:[0,\infty)\rightarrow \mathbb{S}$. Let $X_t(\omega) = \omega(t)$ denote a continuous Markov chain.
Liggett states in proposition 2.29 that
Suppose $X(t)$ is a Markov chain, and let $\tau = \inf\{t\geq 0:X_t\ne X_0\}$ be the time of the first jump. Then $$P^x(\tau>t) = e^{-c(x)t}$$ for some $c(x)\in [0,\infty]$.
So the problem is that if $c(x) = \infty$, then $P^x(\tau > t)=0$ for all $t>0$. Hence, if we take $t\downarrow 0$, then $P^x(\tau > 0)=0$. But, by right continuity, we have $P^x(\tau > 0)=1$ and we reach a contradiction. Is there something wrong with my logic?