Edit: I originally asked if $\phi$ is a closed map. Clearly it isn't in general, as the real numbers can be mapped to the torus with dense image.
Let $G$ and $H$ be connected locally compact groups, and let $\phi: G \rightarrow H$ be a continuous homomorphism. How much can $\phi(G)$ differ from its closure in the topology of $H$? Specifically, is $\phi(G)$ normal in $\overline{\phi(G)}$? If $\phi(G)$ is normal in $\overline{\phi(G)}$, is $\overline{\phi(G)}/\phi(G)$ soluble (abelian?)?