My previous question was specifically concerned with the identity function being the only order-, metric- and addition-preserving function on $\Bbb R$. With this question, I seek answers to the following:
1.Examples of addition-preserving functions on $\Bbb R$
2.Examples of metric-preserving functions on $\Bbb R$
3.Examples of functions on $\Bbb R$ which are both metric- and addition-preserving.
I failed to find useful references/examples on the internet, so I would appreciate references/further readings to be included within the answer(s).
Addition-preserving: $f(x+y) = f(x)+f(y)$. If $f$ is continuous (or measurable, or locally bounded), then such $f$ must be of the form $f(x) = ax$. But there are other solutions, extremely badly behaved, if you believe the Axiom of Choice.
Metric-preserving: $|f(x) - f(y)| = |x-y|$. Such a map must be of the form $f(x) = ax+b$ where either $a=1$ or $a=-1$. This generalizes to Euclidean geometry. A metric-preserving map $f : \mathbb R^n \to \mathbb R^n$ is of the form $f(x) = Ax+b$ where $A$ is an $n \times n$ orthonormal matrix, and $b \in \mathbb R^n$.
Both metric and addition preserving: $f(x) = ax$ where either $a=1$ or $a=-1$.