Representation of $\mathbb{R}$, drop continuity assumption, Axiom of Choice.

Question

A representation, for instance, of $\mathbb{R}$ is a group homomorphism $f: \mathbb{R} \to \text{GL}_n(\mathbb{R})$. If we assume that $f$ is continuous, then there is a very nice formula for all such homomorphisms, namely, that $f$ has to have the form $f(t) = e^{t \cdot a}$ for some fixed matrix $a \in \text{M}_n(\mathbb{R})$. My question is, what happens if we drop the assumption that the map is continuous? Do we have to utilize the Axiom of Choice to construct the objects we are interested in here?

Answer 1

Let $f$ be a discontinuous solution of Cauchy equation. That is, pick a discontinuous additive function $f$ on the reals. Then the function $g(t) = e^{If(t)}$ is a homomorphism between the additive group $\Bbb{R}$ and the multiplicative group $\mathrm{GL}(\Bbb{R}, n)$, where $I$ is the identity matrix. Existence of discontinuous $f$ needs the axiom of choice.