Derp -- perhaps I could have read John's comment 11, or also the Wikipedia article: measurable functions are most definitely out. I was hoping to come up with something almost, but not quite, like the identity function...but it seems it is too special for that.

For anyone still interested in a pathological additive function, there is an [article on 'monster' functions]( linked to at the bottom of the Wikipedia article on [Cauchy's functional equation](