Rational-linear map ($f(qx)=qf(x)(q\in \mathbb{Q})$)

A rational-linear map is a linear map when its domain and codomain are regarded as vector spaces over the rational numbers. An additive map on a rational vector subspace of the real numbers is automatically rational-linear: integer homogeneity follows by repeated addition, and division by a positive integer gives rational homogeneity of an additive function.

Let $D$ be a rational vector subspace of the real numbers and let $t\notin D$. Every element of $D+\mathbb{Q}t$ has a unique expression $d+qt$. For any prescribed real number $a$, the rule $h(d+qt)=f(d)+qa$ extends a rational-linear map $f$ on $D$ and remains rational-linear. The uniqueness follows because a nonzero rational multiple of $t$ cannot lie in $D$.