William wrote:

> I notice: The identity function f(x) = x is order-preserving in all three cases. I think this is unique.

Nice! You mean it's the only one that's order-preserving, addition-preserving and metric-preserving. It's neat how you got examples of functions that preserve two but not all three of these structures.

It takes some work, but one can show that the only order-preserving and addition-preserving functions

$$ f: \mathbb{R} \to \mathbb{R} $$

are of the form

$$ f(x) = c x $$

for some \\(c \ge 0\\). The only one of these functions that's also metric-preserving is the one with \\(c = 1\\).

> Also, it looks like the addition-preserving endomorphisms are fully determined by the value of \\(f(1)\\).

That's very plausible. Surprisingly, it's not true if you believe in the Axiom of Choice, the way most mathematicians do!

It's pretty easy to show that if \\(f\\) is addition-preserving, \\(f(1) = c\\) implies

$$f(x) = c x $$

when \\(x\\) is _rational_. In theory you could cleverly choose \\(f\\) to behave in some other way when \\(x\\) is irrational. And using the Axiom of Choice, you can. But if you require \\(f\\) to be continuous or even just "measurable", the only option is to take \\(f(x) = c x\\) for all \\(x \in \mathbb{R}\\). An order-preserving function must be measurable, so the the only order-preserving and addition-preserving functions

$$ f: \mathbb{R} \to \mathbb{R} $$

are of the form

$$ f(x) = c x $$

for some \\(a \ge 0\\).

For more on this, see:

* Wikipedia, [Cauchy's functional equation](https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation).

In 1821 Cauchy proved that any _continuous_ addition-preserving function \\(f: \mathbb{R} \to \mathbb{R} \\) is of the form \\(f(x) = a x \\). The trick is to show that \\(f\\) must be like this when \\(x\\) is rational. But this was just the start of a long and interesting story!