Question 28
Let $I \subset \mathbb{R}$ be an interval and $f : I \to \mathbb{R}$ be a continuous function. Prove that the following properties are equivalent:
i) $f(x) - f(y) \lt |x - y|$, for every $x, y \in I$;
ii) $\int_x^y f(t)dt - f(x)(y-x) \le \frac{1}{2}(x-y)^2$, for every $x, y \in I$.
i) $f(x) - f(y) \lt |x - y|$, for every $x, y \in I$;
ii) $\int_x^y f(t)dt - f(x)(y-x) \le \frac{1}{2}(x-y)^2$, for every $x, y \in I$.