Semigroup of operators: weak continuity at 0+ implies weak continuity at any t > 0

Question

Let ($E$, $d$) be a metric space. Consider the semigroup $\{P(t)\}_{t\geq 0}$ of bounded linear operators on the Banach space $\hat{C}(E)$ of continuous real functions on ($E$, $d$) vanishing at infinity. Given the following: \begin{equation} \lim_{t \downarrow 0} P_{t}f(x) = f(x), \quad \forall f \in \hat{C}(E), \forall x \in E \end{equation} How do we show that, for any $t>0$, \begin{equation} \lim_{h\downarrow 0} P_{t-h}f(x) = P_{t}f(x), \quad \forall f \in \hat{C}(E), \forall x \in E \end{equation} That is, left pointwise continuity. (I was able to show right pointwise continuity, simply by using the semigroup property.)