I'm doing some reading out of Rudin's Fourier Analysis on Groups and Folland's Abstract Harmonic Analysis for research. Particularly I am on characters and dual groups and I've noticed that neither author delves into any detail about discontinuous characters on locally compact abelian groups (or even gives mention of an example). Clearly such characters exist else we would not look specifically at continuous characters when discussing the dual group.
I tried to come up with an example in the case of $G = (\mathbb R,+)$ for the last hour or so but have gotten nowhere. I considered trying to define the character to be $1$ on dyadic numbers and $-1$ on the complement; I tried looking at an irrational translation of the rationals and letting the character be $1$ on this and $-1$ on the complement; and I even looking at $\mathbb R$ as a vector space over $\mathbb Q$ and trying to define characters via this approach somehow but things got hopelessly complicated at this stage.
Unsurprisingly, got absolutely nowhere in any of these. I get the feeling that the last one is probably relatively close to the kind of methodology one would need to define discontinuous characters (I'm guessing there are some pretty subtle set-theoretic undertones to such a problem given how challenging this has been). Can anyone shed light on this problem in any way? Can such a character be constructed or an algorithm/program be given to find one?