I start with some known preliminaries on the problem:

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

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

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

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 $C\subseteq \mathbf{R}^2$ is a set for which $\alpha x+\beta y \in C$ whenever $\alpha,\beta \in \mathbf{R}^+$ and $x,y \in C$.

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

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?