Keith: I said the proof that _not all_ addition-preserving functions from the reals to the reals uses the axiom of choice. See this:

* Wikipedia, [Cauchy's functional equation](https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation).

In some constructive variants of math all functions from the reals to the reals are continuous. All continuous addition-preserving functions from the reals to the reals are of the form \$$f(x) = c x\$$ for some \$$c \in \mathbb{R}\$$; that's pretty easy to see.