Cauchy's functional equation

Cauchy's functional equation is the functional equation

Solutions to this are called additive functions. Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely ${\displaystyle f:x\mapsto cx}$ for any rational constant ${\displaystyle c}$. Over the real numbers, ${\displaystyle f:x\mapsto cx}$, now with ${\displaystyle c}$ an arbitrary real constant, is likewise a family of solutions; however there can exist other solutions that are extremely complicated. However, any of a number of regularity conditions, some of them quite weak, will preclude the existence of these pathological solutions. For example, an additive function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ is linear if:

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