Cauchy Functional Equation

The Cauchy Functional Equation is the functional equation \[f(x+y) = f(x) + f(y).\]

As with any functional equation, one may attempt to solve this equation for various different choices of the domain of $f$.

Rational Case

If $f: \mathbb Q \to \mathbb Q$ is a function whose domain and range are the rational numbers (or any subset of $\mathbb Q$ closed under addition, like $\mathbb Z$ or $\mathbb N$), the solutions are only the linear functions $f(x)=ax$, with $a\in\mathbb Q$.

Real Case

If $f:\mathbb R \to \mathbb R$ is a function whose domain and range are the real numbers, then there exist solutions to the Cauchy Functional Equation other than linear functions. (To prove this requires some form of the Axiom of Choice, e.g., the existence of a Hamel basis for $\mathbb R$ over $\mathbb Q$. As a result, these functions are "pathological" -- in particular, it is not possible to write down a formula for any such function, and the graph of any such function is dense in the plane.)

However, there are a variety of simple "regularity conditions" such that if $f: \mathbb R \to \mathbb R$ satisfies one of these conditions and the Cauchy Functional Equation, then in must be of the form $f(x) = ax$ for some $a \in \mathbb R$. Examples of such conditions are that $f$ is continuous (or even just continuous at a single point), that $f$ is monotonic, or that $f(x)>0$ for all $x>0$.

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