Give some examples of a general function $f(a,b)$ with $b > 0$ satisfying the following two properties:
- $f(a,b) > a$;
- $f(a_1, b) - f(a_2, b) \leq a_1 - a_2$.
Obviously $f(a, b) = a + g(b)$ with $g(b) > 0$ satisfies those two properties. Are there any other examples except for $f(a, b) = a + g(b)$?
No, there are no other examples.
If (2) is true for all $a_1$ and $a_2$, then it must be true with $a_1$ and $a_2$ interchanged, i.e. $$ f(a_2,b) - f(a_1,b) \le a_2 - a_1$$ Combined with (2), this says $f(a_1, b) - f(a_2, b) = a_1 - a_2$, and thus $f(a,b) - a$ is the same for all $a$. If we let $f(a,b) - a = g(b)$, that says $f(a,b) = a + g(b)$.