Question 69
On the set $A = [0, \infty)$, of all nonnegative real numbers, we consider three functions $f, g, h : A \to A$ and the binary operation $*: A \times A \to A$, defined by
$$
x * y = f(x) + g(y) + h(x) \cdot |x - y|, \quad \text{for any } x, y \ge 0.
$$
If $(A, *)$ is a commutative monoid:
a) show that the function $h$ is continuous on $A$;
b) determine the functions $f, g, h$.
$$
x * y = f(x) + g(y) + h(x) \cdot |x - y|, \quad \text{for any } x, y \ge 0.
$$
If $(A, *)$ is a commutative monoid:
a) show that the function $h$ is continuous on $A$;
b) determine the functions $f, g, h$.