Cauchy's functional equation over the rationals

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Find all functions $f : Q \to Q$ such that $f (x + y) = f (x) + f (y)$ for all rational $x$ and $y$ .

Posed by Augustin-Louis Cauchy, 1821

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algebra functional-equations

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$f (x) = c x$ for a rational constant $c$ .

History and references

Cauchy studied the equation in his Cours d'analyse of 1821. Over the rationals the answer is forced, $f (x) = c x$ with $c = f (1)$ . Over the reals it is forced as well once $f$ is continuous, monotone or bounded on some interval, but with the axiom of choice there are other solutions, whose graphs are dense in the plane.

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Cauchy's functional equation

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