Does there exist a continuous map $f : [0, 1] \to [0, 1] \times [0, 1]$ such that the pre-image of any point of the square $[0, 1] \times [0, 1]$ contains precisely two points of the interval $[0, 1]$?
I guess the answer is no, but i have no idea how to consider this, even how to start? Anyone can help? Thanks!