Is Feller semigroup in this definition strongly continuous?

Question

Let $C_0=C_0(\mathbb R^d)$ be the space of continuous functions vanishing at infinity, equipped with the usual supremum norm $\| \cdot \|$. Let $T=(T_t)_{t\geq 0}$ be a semigroup of operators satisfying

1. $T_0 = I$ and $T_{t+s} = T_tT_s$ and contraction property: $\| T_t f \| \leq \|f\|$

2. $\forall t \geq 0 $, $T_tC_0 \subset C_0$

3. $\forall x \in \mathbb R^d,\forall f \in C_0$, we have $T_tf(x) \rightarrow f(x)$ as $t \downarrow 0$

Can we conclude that $\| T_t f -f\| \rightarrow 0$ as $t \downarrow 0$, i.e., $T$ is strongly continuous at 0?

This is the Remark just after Definition 27.3 this lecture notes. My motivation is to verify another question of mine here. For now, I don't see why this is true. Thank you in advance.

Answer 1

It is a well-known result which appears in many textbooks. For example, you may refer to the book "continuous martingales and Brownian motion" written by Revuz and Yor, chapter III, Proposition 2.4.