At dinner the other night, our table wobbled. While the waitress fetched a wedge to stabilize it, I told my fellow diners about an applicable mathematical proof: If the legs of a table are even, but the table wobbles because of an uneven floor surface, then rotating the table will always find a position where all four legs are touching the ground.
They were skeptical, but it is true. The ground must be continuous, in the theory of functions sense: no cliff-like changes in height. As proof, here's an intuitive thought experiment, and the same concept, but full of mind-bending rigor which I don't understand.