The linearity proof given above also applies to any set\\ alpha\\ mathbb, a scaled copy of the rationals, showing that the only solutions are linear when the domain of f is restricted to such sets.
The linearity proof given above also applies to f:\\ alpha\\ mathbb\\ to\\ mathbb, where\\ alpha\\ mathbb is a scaled copy of the rationals. This shows that the only solutions are linear when the domain of f is restricted to such sets.