Question 46
A function $f: \mathbb{R} \to \mathbb{R}$ is said to have property ($\mathcal{P}$) if for any sequence of reals $(x_n)_{n \ge 1}$, such that $(f(x_n))_{n \ge 1}$ is convergent, the sequence $(x_n)_{n \ge 1}$ is also convergent. Prove that a surjective function that possesses property ($\mathcal{P}$) is continuous.
Mihai Piticari and Sorin Rădulescu
Mihai Piticari and Sorin Rădulescu