Suppose $G$ is a locally compact abelian (LCA) group and
$f: H \rightarrow G$
a continuous group homomorphism, where $H$ is a topological torsion group. Does it follow that the topological closure of the image in $G$ is again topological torsion?
(Edit: A group is $\textit{topologically torsion}$ if $\operatorname{lim}_n n! x = 0$ for all $x$ in the group, or equivalently (by a theorem of Robertson) if every character on the group takes values in Q/Z (as opposed to R/Z), or equivalently if both $G$ and its Pontryagin dual are simultaneously totally disconnected)
No. For instance, let $H$ be the discrete group $\mathbb{Q}/\mathbb{Z}$, $G=\mathbb{R}/\mathbb{Z}$, and let $f$ be the natural inclusion map. Then the image of $f$ is dense in $G$, but $G$ is not topologically torsion.