Stability of group homomorphisms in the compact-open topology

We introduce the notion of stability of continuous homomorphisms of topological groups with respect to the compact-open topology on the space of all continuous functions between them and prove that the characters of any locally compact abelian group are stable in this sense. Using this notion we also generalize a global stability result on continuous homomorphisms between compact groups to a local version for continuous homomorphisms from any locally compact group to an arbitrary topological group and show the necessity of the assumptions of the theorem through some counterexamples. Finally, we separately discuss the special cases of metrizable and discrete groups, respectively. As an application of the latter we prove a purely algebraic result on extendability of finite partial functions to group homomorphisms.