APMO 2026 Problem 3

Let $\mathbb{R}_+$ denote the set of positive real numbers. Determine all the functions $f : \mathbb{R}_+ \to \mathbb{R}$ such that for all $x, y, z \in \mathbb{R}_+$,

$$|x - y| < |y - z| \text{ if and only if } |f(x) - f(y)| < |f(y) - f(z)|.$$