Pontryagin Duality for locally compact abliean groups gives plenty of continuous (unitary) characters $\chi : A \to \mathbb{R} / \mathbb{Z}$, but if we do not assume local compactness, can anything be said? In particular, is the following true?

Every abelian Hausdorff topological group has a nontrivial continuous (unitary) character

I believe this to be false, but cannot find a concrete example.