An additive $f : \mathbb{R} \to \mathbb{R}$ satisfies $f(0) = 0$, $f(-x) = -f(x)$ and $f(qx) = q\,f(x)$ for every rational $q$ and every real $x$; in particular $f(q) = q\,f(1)$ at every rational $q$

Let $f : \mathbb{R} \to \mathbb{R}$ be additive (Cauchy's functional equation $f(x+y) = f(x) + f(y)$, and the additive functions $\mathbb{R} \to \mathbb{R}$), and identify $\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}$ along the canonical embeddings (The naturals embed in the integers, The integers embed in the rationals, The rationals embed densely in the reals), writing $\iota(n)$ for the canonical natural of $n$ in $\mathbb{R}$ (The canonical natural $\iota(n) = n \cdot 1_F$ of a field). Then, for every real $x$:

1. $f(0) = 0$;
2. $f(-x) = -f(x)$;
3. $f(\iota(n)\,x) = \iota(n)\,f(x)$ for every $n \in \mathbb{N}$;
4. $f(m x) = m\,f(x)$ for every integer $m$;
5. $f(q x) = q\,f(x)$ for every rational $q$.

In particular, taking $x = 1$ in claim 5, $f(q) = q\,f(1)$ at every rational $q$: an additive function is determined on $\mathbb{Q}$ by its value at $1$.

What this does not say. Claim 5 is $\mathbb{Q}$-homogeneity, not $\mathbb{R}$-homogeneity: nothing here gives $f(\lambda x) = \lambda f(x)$ for irrational $\lambda$, and that is exactly the gap that FALSE: every additive $f : \mathbb{R} \to \mathbb{R}$ is of the form $x \mapsto cx$ for a single real $c$ shows cannot be closed without a regularity hypothesis.

• The induction is on $\mathbb{N}$ and everything else is algebra. Only claim 3 needs induction; claims 4 and 5 are obtained from it by the two field operations, and claims 1 and 2 are two substitutions into the equation. The base case is $n = 0$, where $\iota(0) = 0$ and the identity reads $f(0) = 0$; it is a genuine case and not a convention, since $\mathbb{N}$ contains $0$.

• This is the whole of the algebraic theory. Every regularity theorem about Cauchy's equation (Six regularity conditions each force an additive $f : \mathbb{R} \to \mathbb{R}$ to be $x \mapsto f(1)x$: continuity at a single point, monotonicity on a nondegenerate interval, boundedness above on one, boundedness below on one, constancy of sign on one, and a graph that is not dense in $\mathbb{R}^{2}$) works by combining claim 5 with density of $\mathbb{Q}$ in $\mathbb{R}$: the value of $f$ is pinned on a dense set, and a regularity hypothesis is what forbids the values off that set from being arbitrary.