I have a question about Feller processes.

In this paper enter link description here 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.

1. For every $f \in C_{\infty}(E)$ and $t >0$, we have $P_{t}f \in C_{\infty}(E)$.
2. 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?

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.

1. For every $f \in C_{\infty}(E)$ and $t >0$, we have $P_{t}f \in C_{\infty}(E)$.
2. 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?

I have a question about Feller processes.

In this paper enter link description here 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.

1. For every $f \in C_{\infty}(E)$ and $t >0$, we have $P_{t}f \in C_{\infty}(E)$.
2. 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*} P_{t}f(x)=E_{x}[f(X_t)] \to 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?

I have a question about Feller processes.

In this paper enter link description here 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.

1. For every $f \in C_{\infty}(E)$ and $t >0$, we have $P_{t}f \in C_{\infty}(E)$.
2. 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?

I have a question about Feller processes.

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.

1. For every $f \in C_{\infty}(E)$ and $t >0$, we have $P_{t}f \in C_{\infty}(E)$.
2. 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*} P_{t}f(x)=E_{x}[f(X_t)] \to 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 we need the condition 2?

I have a question about Feller processes.

In this paper enter link description here 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.

1. For every $f \in C_{\infty}(E)$ and $t >0$, we have $P_{t}f \in C_{\infty}(E)$.
2. 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*} P_{t}f(x)=E_{x}[f(X_t)] \to 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?