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 $g : S \to H$ be a (semigroup) homomorphism, then there exists a unique (group) 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.

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.

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 $g : S \to H$ be a (semigroup) homomorphism, then there exists a unique (group) homomorphism $f : G \to H$ whose restriction to $S$ is $g$.

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 $g : S \to H$ be a (semigroup) homomorphism, then there exists a unique (group) 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.