If $ f: \mathbb{R} \to \mathbb{R}$ is a continuous surjection, must it be open?

I think not. I proved if $ f: \mathbb{R} \to \mathbb{R}$ is an open continuous surjection, then $ f$ is a homeomorphism. So, if the question is true, every continuous surjection must be a homeomorphism. But, I didn’t find a counterexample. Can someone help me?