  We prove that arbitrary homomorphisms from one of the groups ${\rm
Homeo}(\ca)$, ${\rm Homeo}(\ca)^\N$, ${\rm Aut}(\Q,<)$, ${\rm Homeo}(\R)$, or
${\rm Homeo}(S^1)$ into a separable group are automatically continuous. This
has consequences for the representations of these groups as discrete groups.
For example, it follows, in combination with a result on V.G. Pestov, that any
action of the discrete group ${\rm Homeo}_+(\R)$ by homeomorphisms on a compact
metric space has a fixed point.
