Question 79
Let $f : \mathbb{R} \to \mathbb{R}$ be a function so that for every $x, y \in \mathbb{R}$, $x \lt y$, there exists $z \in (x, y)$ so that
$$
(y - x)f(z) \le (y - z)f(x) + (z - x)f(y).
$$
a) Give an example of a non-convex function $f$ with the given property.
b) Prove that a continuous function with the given property is convex.
$$
(y - x)f(z) \le (y - z)f(x) + (z - x)f(y).
$$
a) Give an example of a non-convex function $f$ with the given property.
b) Prove that a continuous function with the given property is convex.