Аннотация:In this note, we establish the continuity of every locally bounded homomorphism of a connected Lie group $G$ into a Lie group without nontrivial compactconnected subgroups. In particular, the assertion holds for locally bounded homomorphisms into connected simply connected solvable Lie groups and for the homomorphisms into the universal covering group of $\operatorname{SL}(2,\Bbb R)$.