RO-NMO 2017 Problem 111

Let be a surjective function $f:\mathbb{R}\longrightarrow\mathbb{R}$ that has the property that if the sequence $\left( f\left( x_n \right) \right)_{n\ge 1}$ is convergent, then the sequence $\left( x_n \right)_{n\ge 1}$ is convergent. Prove that it is continuous.