In the last few days I've been thinking on and off about these two problems and I can't get my head around them:
Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuous open map.
If $f$ is surjective and $\varnothing \neq A \subseteq \mathbb{R}$ is an open set, can $f^{-1}(A)$ be bounded?
Can $f$ be also closed?
The only progress I managed to make is saying that in 2. since $\mathbb{R}^2$ is both open and closed, then its image should be both open and closed, thus it must be that $f$ is surjective, since $\mathbb{R}$ is connected.
