I came across this exercise, the main problem for me are the restrictions, i need to find examples for maps $f: \mathbb {R}^2 \rightarrow \mathbb {R}^2$ or subsets of them such that $f$ is only one or two of the three at the same time. (I.e $f$ is open but no closed nor continuous)
But i don't know How to build such examples, as i also don't see the kind of patologies i should be looking for