real analysis - Is it useful to know that automorphisms on $(mathbb R^{gt0},+)$ are always continuous?

I find it interesting that any automorphism of the semigroup $(\mathbb R^{\gt0},+)$ is continuous.

This is also true if we assume the Axiom of Choice; c.f. Automorphisms on (R,+) and the Axiom of Choice. The simple argument is that any morphism on $(\mathbb R^{\gt0},+)$ must preserve the order, and if you can show that it must be surjective, then it has no gaps and must be continuous; see this.

Has this found any use in the exposition of mathematical theories?

Also interested in any answer that shows two ways of proving something, one proof long and laborious, and the second argument, using this fact, considerably shorter, albeit more abstract.

The very best answers would be those that employ the theory of magnitudes.