Thread: How to prove a finite number of parabolas and their inside can't cover a whole plane?

Originally Posted by OliverQ

Maybe putting this question here is not so proper, but maybe I can get the answers I need here. How can I prove a finite number of parabolas and their inside(the side which contains the focus) can't cover a whole plane? I know they can't, but I can't prove it...
Help!

Here is a way to procede: what can be said about the intersection of a parabola and a line? You can easily prove that any line that is not parallel to the axis has a bounded intersection (a line segment, a point or the empty set) with the parabola and its interior. If you're given a finite number of parabolas, you can choose a line that is parallel to none of the axes, hence it has only a bounded intersection with all the parabolas and their interiors.