1. \\(f(x) = e^x\\) is order-preserving, but \\(f(x) = e^{-x}\\) is not.

2. \\(f(x) = -x - 1\\) is metric-preserving, but \\(f(x) = 0\\) is not.

3. \\(f(x) = x\|I_Q(x) - 1\|\\) where \\(I_Q(x)\\) is the [Dirichlet function](https://en.wikipedia.org/wiki/Nowhere_continuous_function#Dirichlet_function) is addition-preserving; it is metric and order-preserving almost everywhere, in the pathological sense noted in comment 11, being the identity function everywhere apart from the rationals where it is zero.

2. \\(f(x) = -x - 1\\) is metric-preserving, but \\(f(x) = 0\\) is not.

3. \\(f(x) = x\|I_Q(x) - 1\|\\) where \\(I_Q(x)\\) is the [Dirichlet function](https://en.wikipedia.org/wiki/Nowhere_continuous_function#Dirichlet_function) is addition-preserving; it is metric and order-preserving almost everywhere, in the pathological sense noted in comment 11, being the identity function everywhere apart from the rationals where it is zero.