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?

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    $\begingroup$ You may want to mention that a solution to the "multidimensional Cauchy equation" is known for the general case of a continuous additive function $f: \mathbf R^n \to \mathbf R$ (Theorem 5.5.2 in the 2nd edition of M. Kuczma's An Introduction to the Theory of Functional Equations and Inequalities), and that much is also known for the restricted Cauchy equation in $\mathbf R^n$ (Section 13.6 in Kuczma's book). $\endgroup$ – Salvo Tringali Mar 7 '17 at 9:18
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    $\begingroup$ For $f$ a map between complex vector spaces the "multidimensional Cauchy equation" is also a special case of Thm 7.2 in Risteski and Covachev's Complex vector functional equations (2002). For more about the solution to the Cauchy matrix functional equation they refer to Gheorghiu, C. R. Acad. Sci. Paris 256, 3562 (1963) and Kuwagaki, J Math Soc Japan 14, 359 (1962). $\endgroup$ – Jules Lamers Mar 7 '17 at 10:12
  • $\begingroup$ I cannot edit my previous comment, but on the last line I should have written "for restricted Cauchy equations in $\mathbf R^n$" (there is nothing like "the restricted Cauchy equation"). $\endgroup$ – Salvo Tringali Mar 7 '17 at 17:21

As I have just learned from Janusz Matkowski, this follows from Theorems 5.5.2 and 18.2.1 in Kuczma's book (the same mentioned in my comments to the OP), after ruling out the trivial case when the cone $C$ is a line or a half-line. In particular, Theorem 5.5.2 is about the (unrestricted) Cauchy functional equation in $\mathbf R^n$, and shows that the only continuous solutions are precisely the functions of the form $\mathbf R^n \to \mathbf R: (x_1, \ldots, x_n) \mapsto \sum_{i=1}^n c_i x_i$ with $c_1, \ldots, c_n \in \mathbf R$, while Theorem 18.2.1 reads as follows:

Let $G$ and $H$ be abelian groups (written additively), and let $S$ be a subsemigroup of $G$ such that $G = S - S := \{x-y: x, y \in S\}$. If $g : S \to H$ is a (semigroup) homomorphism, then there exists a unique homomorphism $f : G \to H$ whose restriction to $S$ is $g$.

Each of these results is somehow a piece of folklore (for people working primarily on functional equations and related topics), but I think it is not harmful to have a reference.

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    $\begingroup$ For the record, Theorem 18.2.1 in Kuczma's book is borrowed from: J. Aczél, J.A. Baker, D.Z̆. Djoković, Pl. Kannappan, and F. Radó, Extensions of certain homomorphisms of subsemigroups to homomorphisms of groups, Aequationes Math. 6 (1971), 263--271. $\endgroup$ – Salvo Tringali Mar 9 '17 at 21:55

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