Consider the map $$T\colon[0,\infty)\to\operatorname{Lin}(C_{b}(\mathbb{R}),C_{b}(\mathbb{R})),\qquad (T(t)f)(x):=f(x+t),$$ where $C_{b}(\mathbb{R})$ is the space of bounded continuous functions from $\mathbb{R}$ to $\mathbb{R}$, endowed with the supremum norm. I am looking for a function $f\in C_{b}(\mathbb{R})$ such that the orbit map $$O_{f}\colon[0,\infty)\to C_{b}(\mathbb{R}),\qquad O_{f}(t):=T(t)f$$ is not continuous.
To find such a function, I considered the sequence $t_{n}:=\tfrac{1}{n}$ in $[0,\infty)$ and tried to find $f\in C_{b}(\mathbb{R})$ such that $$\|O_{f}(0)-O_{f}(t_{n})\|_{\infty}=\sup_{x\in\mathbb{R}}|f(x)-f(x+\tfrac{1}{n})|\not\to0\qquad\text{as}\qquad n\to\infty.$$
The trivial examples, such as constant functions obviously do not work. Then I consider (bounded continuous) functions that behave like $\sqrt[3]{x}$ around $x=0$, because its slope becomes infinitely large, but this also doesn't seem to work.
Any suggestions would be greatly appreciated.