On necessity of Feller property I have a question about Feller processes.
In this paper "On the doubly Feller property of resolvent" the transition semigroup $(P_t)_{t \ge 0}$ of a Hunt process (on a metric space $E$) is said to have the Feller property if the following two conditions are satisfied. 


*

*For every $f \in C_{\infty}(E)$ and $t >0$, we have $P_{t}f \in C_{\infty}(E)$.

*For every $f \in C_{\infty}(E)$ and $x \in E$, $\lim_{t \to 0}P_{t}f(x)=f(x)$.


Here, $C_{\infty}(E)$ is the family of continuous functions on $E$ vanishing at infinity.
My question
I think the transition semigroup $(P_t)_{t \ge 0}$ of a Hunt process $X$ always satisfies the condition 2. Since $f(X_t) \to f(X_0)$ as $t \to 0$ from the right continuity of the sample path of $X$. This implies
\begin{align*}
\lim_{t \to 0}P_{t}f(x)=\lim_{t \to 0}E_{x}[f(X_t)]=E_{x}[f(X_0)]=E_{x}[f(x)]=f(x)
\end{align*}
by the dominated convergence theorem and the normal property of $X$.
Why they need the condition 2?
 A: Sorry I probably missed "$(P_t)_{t \ge 0}$ of a Hunt process" when I first composed this answer and that is why confusion arise. 
I think it is most beneficial if we start from and stick to definition.
Hunt process is a strong Markov process with some regularity on sample functions, usually càdlàg sample functions but subject to some alternations. Suppose $X$ is on a (locally) compact space $E$ and there is a one-to-one correspondence between Hunt process and Feller-Dynkin semigroups. 
The paper you cited provided (1)(2) as a set of equivalent conditions that allows you to regard the transition semigroup $(P_t)_{t \ge 0}$ as a Feller-Dynkin semigroup as well. A Hunt process may admits a transition semigroup satisfying (2) of course but it also needs (1) to make the operations within the transition semingroup closed.
I hope I made myself clear now.
A: The Feller property is sufficient (bot not necessary) for the existence of a Hunt process associated with $(P_t)$. As you observe, condition (2)  in the Feller property is (in isolation) also a necessary consequence of the Hunt property. 
