Show that $S$ is continuous on $\mathbf{R}_{+}$, where $S(t) = \operatorname{Ent}_{\varphi}\left(P_{t}(f)\right)$. Hint: You may first show that, if $x \in \mathbf{R}$, then $t \mapsto P_{t}(f)(x)$ is continuous on $\mathbf{R}_{+}$.
Show that $S$ is continuous on $\mathbf{R}_{+}$, where $S(t) = \operatorname{Ent}_{\varphi}\left(P_{t}(f)\right)$.
Hint: You may first show that, if $x \in \mathbf{R}$, then $t \mapsto P_{t}(f)(x)$ is continuous on $\mathbf{R}_{+}$.