  We establish a necessary and sufficient condition for an action of a lattice
by homeomorphisms of the circle to extend continuously to the ambient locally
compact group. This condition is expressed in terms of the real bounded Euler
class of this action. Combined with classical vanishing theorems in bounded
cohomology, one recovers rigidity results of Ghys, Witte--Zimmer, Navas and
Bader--Furman--Shaker in a unified manner.
