On page 163, section 11, chapter V, Conway's functional analysis.
Let $G$ be a locally compact group and $f\in\mathrm{C}_b(G)$. We say $f$ is almost periodic if \begin{equation} \mathcal{O}_f\,\colon\!=\text{closure of }\left\{f_x\in\mathrm{C}_b(G)\colon x\in G\right\} \end{equation} is compact in $\mathrm{C}_b(G)$, write $f\in\mathrm{AP}(G)$.
My question: I want to show if $G$ is not compact, then the only function in $\mathrm{AP}(G)$ having compact support is the zero function.
But I have no idea.