Let $\displaystyle f: [0,1] \rightarrow \mathbb{R} $ be a function with Darboux's property. $\displaystyle \forall \ y \in \mathbb{R}$, we have $\displaystyle f^-1({{y}}) $ is closed. Prove that $\displaystyle f$ is continuous.

So I have an idea of how to approach this, but I'm not entirely sure on the closed part - does that necessary entail that the function is surjective? (If not, the pre-image wouldn't exist?)