Here’s a nice, light application of the Intermediate Value Theorem we came up with yesterday. Let’s say you’ve got a square table with four legs at the corners. They’re all the same length, but the floor is warped and so the table wobbles. As it happens, you only need to rotate the table around its center to find a position with all four legs touching the ground.
Put three of the table’s legs on the floor so the fourth either hangs in the air or presses down into the ground. As we rotate the table, the distance this leg is from the floor is a real-valued function of the angle we’ve turned the table from its starting point. If the leg starts above the floor, then as we rotate the table a quarter-turn it must push down into the floor. To see this, notice that instead of rotating a quarter-turn we could just push that leg down to the floor. As we do, the next leg over has to get pushed into the floor. So our function will be positive for some angles and negative for others. Now since the height of the leg is a continuous function of the angle, and the set of angles is connected, the IVT tells us that there must be some angle where the height of the leg is zero — it exactly touches the floor along with the other three legs.
The details get a bit hairier than this rough sketch. For more about them, and about other shapes of tables, you can read this article on the arXiv. But the basic idea comes down to the IVT.