Ok, we've warmed up enough! Now for the real thing:

Let and be topological groups, being connected as a topological space, and let be a homomorphism of topological groups, i.e.: Continuous, and: (or , in simplified notation) . Suppose moreover that , i.e.: such that: . Then , i.e.: is onto.

Can anyone suggest a proof? Then I will explain why I picked this problem up.