I start with some known preliminaries on the problem:

**Classical result.** The one-dimensional Cauchy functional equation
$$
\forall x,y \in \mathbb{R}, \,\,\,f(x+y)=f(x)+f(y)
$$
with $f:\mathbb{R}\to \mathbb{R}$ is only solved by the trivial solutions $f(x)=cx$, for some $c \in \mathbb{R}$, if $f$ satisfies for some additional conditions, e.g., continuity.


**Classical result with restricted domain.** Now let $\mathbb{R}^+:=(0,\infty)$. It is clear from the proof of the above classical result that if $f:\mathbb{R}^+\to \mathbb{R}^+$ is a continuous function such that
$$
\forall x,y \in \mathbb{R}^+, \,\,\,f(x+y)=f(x)+f(y) \, ,
$$
then there exists $c \in \mathbb{R}^+$ such that $f(x)=cx$ for all $x$.


**Multidimensional Cauchy functional equation.** It is also well known that if $f:\mathbb{R}^2\to \mathbb{R}$ is a continuous function such that
$$
\forall x,y \in \mathbb{R}^2, \,\,\,f(x+y)=f(x)+f(y),
$$
then there exist $A,B \in \mathbb{R}$ such that $f(x,y)=Ax+By$ for all $(x,y) \in \mathbb{R}^2$.

I know that the following generalization holds true as well. In particular, I already know how to prove it, by using a variant of the classical proof. In the following, a cone $C\subseteq \mathbb{R}^2$ is a set for which $\alpha x+\beta y \in C$ whenever $\alpha,\beta \in \mathbb{R}^+$ and $x,y \in C$.

>**Fact.**  Let $C\subseteq \mathbb{R}^2$ be a non-empty cone and $f:C \to \mathbb{R}$ be a continuous function such that
$$
\forall x,y \in C, \,\,\,f(x+y)=f(x)+f(y).
$$
Then there exist $A,B \in \mathbb{R}$ such that $f(x,y)=Ax+By$ for all $(x,y) \in C$.

Is it a known result? In such case, does anyone have a reference for this result?