Cauchy's functional equation

Cauchy's functional equation is the functional equation: $${\displaystyle f(x+y)=f(x)+f(y).\ }$$

A function ${\displaystyle f}$ that solves this equation is called an additive function. Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely ${\displaystyle f\colon x\mapsto cx}$ for any rational constant ${\displaystyle c.}$ Over the real numbers, the family of linear maps ${\displaystyle f:x\mapsto cx,}$ now with ${\displaystyle c}$ an arbitrary real constant, is likewise a family of solutions; however there can exist other solutions not of this form 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 ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ is linear if:

• ${\displaystyle f}$ is continuous (Cauchy, 1821). In fact, it suffices for ${\displaystyle f}$ to be continuous at one point (Darboux, 1875).
• ${\displaystyle f(x)\geq 0}$ or ${\displaystyle f(x)\leq 0}$ for all ${\displaystyle x\geq 0}$.
• ${\displaystyle f}$ is monotonic on any interval.
• ${\displaystyle f}$ is bounded on any interval.
• ${\displaystyle f}$ is Lebesgue measurable.
• ${\displaystyle f\left(x^{n+1}\right)=x^{n}f(x)}$ for all real ${\displaystyle x}$ and some positive integer ${\displaystyle n}$.

On the other hand, if no further conditions are imposed on ${\displaystyle f,}$ then (assuming the axiom of choice) there are infinitely many other functions that satisfy the equation. This was proved in 1905 by Georg Hamel using Hamel bases. Such functions are sometimes called Hamel functions.^[1]

The fifth problem on Hilbert's list is a generalisation of this equation. Functions where there exists a real number ${\displaystyle c}$ such that ${\displaystyle f(cx)\neq cf(x)}$ are known as Cauchy-Hamel functions and are used in Dehn-Hadwiger invariants which are used in the extension of Hilbert's third problem from 3D to higher dimensions.^[2]

This equation is sometimes referred to as Cauchy's additive functional equation to distinguish it from Cauchy's exponential functional equation ${\displaystyle f(x+y)=f(x)f(y),}$ Cauchy's logarithmic functional equation ${\displaystyle f(xy)=f(x)+f(y),}$ and Cauchy's multiplicative functional equation ${\displaystyle f(xy)=f(x)f(y).}$