Max and Mike both mentioned that there are addition-preserving self-maps on the reals that are unbounded on every non-trivial interval. But such examples require the axiom of choice. In a different context I happened to mention that just two months ago: Do we need the axiom of choice? If, instead, we add the axiom of measurability, then the counterexamples disappear: let f be a measurable midpoint-preserving map from R to R; we can easily specialize to the case that f0 = 0, hence f is a measurable group endomorphism; for any real a consider the induced group homomorphism from R/aZ to R/(fa)Z; any measurable group character is continuous and that's enough to force f to be continuous. (28 Jan) The example that Michael Wendt and I described (well, almost the same example) we didn't need choice but we did need -- as all good toponomers know -- the law of excluded middle for the equality predicate on the reals (in the category of spatial sheaves over a perfect space it is the case that all real-valued maps from a closed interval are, indeed, bounded.)