Left translation on $C_b (\mathbb{R})$ is not a strongly continous semigroup

Question

I'm studying strongly continuous semigroups right now with the book One-Parameter Semigroups for Linear Evolution Equations by Engel, Nagel. In this book there is a small section about the left translation operator which is defined for a function $f\colon \mathbb{R} \to \mathbb{C}$ and $t\geq 0$ by
\begin{equation*}
(T(t)f)(s):=f(s+t), \ \ s \in \mathbb{R}
\end{equation*}
and it was claimed that the operatorfamily $(T(t))_{t\geq 0}$ does not form a strongly continous semigroup if the underlying function space for $f$ is $(C_b(\mathbb{R}),||\cdot||_{\infty})$ the space of bounded continuous functions on $\mathbb{R}$. So one has to find a function $f\in C_b(\mathbb{R})$ such that $\lim_{t\downarrow 0}(T(t)f)(s)=f(s)$ does not hold and therefore the strong continuity is not fullfilled but I couldn't think of an example. My guess was to approximate such an $f$ by a sequence $(f_n)_{n\in \mathbb{N}}\subset C_b(\mathbb{R})$ which converges pointwise but not uniformly to $f$.

Answer 1

HINT:

Take $f(x) = \sin (x^2)$.