Question 22
Prove that, for a function $f : \mathbb{R} \to \mathbb{R}$, the following affirmations are equivalent:
(i) $f$ is differentiable on $\mathbb{R}$ with continuous derivative;
(ii) for any $a \in \mathbb{R}$ and two sequences $(x_n)_{n \ge 1}$ and $(y_n)_{n \ge 1}$, with limit $a$, such that $x_n \neq y_n$, for all $n \in \mathbb{N}^*$, the sequence $\left( \frac{f(x_n) - f(y_n)}{x_n - y_n} \right)_{n \ge 1}$ is convergent.
(i) $f$ is differentiable on $\mathbb{R}$ with continuous derivative;
(ii) for any $a \in \mathbb{R}$ and two sequences $(x_n)_{n \ge 1}$ and $(y_n)_{n \ge 1}$, with limit $a$, such that $x_n \neq y_n$, for all $n \in \mathbb{N}^*$, the sequence $\left( \frac{f(x_n) - f(y_n)}{x_n - y_n} \right)_{n \ge 1}$ is convergent.