  We prove that a group homomorphism $\varphi\colon L\to G$ from a locally
compact Hausdorff group $L$ into a discrete group $G$ either is continuous, or
there exists a normal open subgroup $N\subseteq L$ such that $\varphi(N)$ is a
torsion group provided that $G$ does not include $\mathbb{Q}$ or the $p$-adic
integers $\mathbb{Z}_p$ or the Pr\"ufer $p$-group $\mathbb{Z}(p^\infty)$ for
any prime $p$ as a subgroup, and if the torsion subgroups of $G$ are small in
the sense that any torsion subgroup of $G$ is artinian. In particular, if
$\varphi$ is surjective and $G$ additionaly does not have non-trivial normal
torsion subgroups, then $\varphi$ is continuous.
  As an application we obtain results concerning the continuity of group
homomorphisms from locally compact Hausdorff groups to many groups from
geometric group theory, in particular to automorphism groups of right-angled
Artin groups and to Helly groups.
