Show that $S$ is continuous on $\mathbf{R}_+$. Hint: You may first show that, if $x \in \mathbf{R}$, $t \mapsto P_t(f)(x)$ is continuous on $\mathbf{R}_+$.
Show that $S$ is continuous on $\mathbf{R}_+$.
\textit{Hint: You may first show that, if $x \in \mathbf{R}$, $t \mapsto P_t(f)(x)$ is continuous on $\mathbf{R}_+$.}