Question 8
Let $I \subset \mathbb{R}$ be an open interval and $f, g : I \to \mathbb{R}$ two functions that satisfy
$$
\frac{f(x) - g(y)}{x - y} + |x - y| \ge 0, \text{ for any } x, y \in I, x \ne y.
$$
i) Deduce that $f$ and $g$ are non-decreasing.
ii) Find $f, g : \mathbb{R} \to \mathbb{R}$, $f \ne g$, having the above property.
$$
\frac{f(x) - g(y)}{x - y} + |x - y| \ge 0, \text{ for any } x, y \in I, x \ne y.
$$
i) Deduce that $f$ and $g$ are non-decreasing.
ii) Find $f, g : \mathbb{R} \to \mathbb{R}$, $f \ne g$, having the above property.