Map from real numbers to real numbers as abelian groups

What is an example of an abelian group homomorphism from $\mathbb{R}$ to $\mathbb{R}$, respecting addition, that is NOT a linear transformation?

I'm trying to understand to what extent the scaling condition for the linear transformation is necessary. For $\mathbb{Q}$ vector spaces, maps that preserve addition also preserves scaling. It seems the situation is more complicated for $\mathbb{R}$. I guess I can "choose" a basis of $\mathbb{R}$ as a $\mathbb{Q}$ vector space and go from there, but are there any explicit examples, preferably without using the Axiom of Choice?

More generally, is there a description of the group of abelian group homomorphisms from $\mathbb{R}^n$ to $\mathbb{R}^m$?

Edit: While the answer below was being typed, the question was changed and now asks for a choiceless example. Recall (Solovay), that if ZF is consistent, then so is $ZF$ plus every set of reals is Lebesgue measurable. Thus it is consistent that the Cauchy functional equation has only linear solutions.

Let $H$ be a (Hamel) basis for the reals over the rationals. Define the function $\varphi$ from $H$ to $H$ by picking two distinct elements $h_1$ and $h_2$ of $H$, and letting $\varphi$ interchange $h_1$ and $h_2$, and leave the other elements of $H$ fixed.

Extend $\varphi$ to a homomorphism from $\mathbb{R}$ to $\mathbb{R}$ by $\mathbb{Q}$-linearity.

For more information, please google Hamel basis and Cauchy functional equation.