Weakly linear partial function ($f(x+y)=f(x)+f(y),q\mathrm{dom}(f)\subseteq \mathrm{dom}(f)(q\in \mathbb{Q})$)

A weakly linear partial function on the real numbers has a domain closed under rational scalar multiplication and addition, and it preserves addition on that domain. A nonempty domain is a vector space over the rational numbers, and additivity implies rational homogeneity of an additive function by integer multiplication and division. The empty function also satisfies the defining conditions vacuously. This terminology does not assert homogeneity under arbitrary real scalars.