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](http://www.cofault.com/2010/01/hunt-for-addictive-monster.html) linked to at the bottom of the Wikipedia article on [Cauchy's functional equation](https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation).

For anyone still interested in a pathological additive function, there is an [article on 'monster' functions](http://www.cofault.com/2010/01/hunt-for-addictive-monster.html) linked to at the bottom of the Wikipedia article on [Cauchy's functional equation](https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation).