Let $X$ be the space of all continuous functions $u \in C(\mathbb R)$ where $$\lim_{t \to \pm \infty}u(t)=0$$ provided with the $\sup$-norm $$\vert\vert u \vert\vert_\infty=\sup_{t \in \mathbb R}\vert u(t)\vert.$$ Now let $g \in C(\mathbb R)$ a bounded, continuous function. For $t \in \mathbb R$ define $$S(t): X \to X, \\ u \mapsto S(t)u :=\exp(tg)u$$ where $S(t)$ is the multiplication operator of the function $\exp(tg)$.
How can I show that $(S(t))_{t \in \mathbb R}$ is a uniformly continuous group of bounded operators?