# Intermediate value theorem/Turn around table/Example

Suppose that a regular quadratic table with four table legs ${\displaystyle {}A,B,C,D}$ is standing on an uneven but stepfree underground. At the moment, it stands on the legs ${\displaystyle {}A,B,C}$, and the leg ${\displaystyle {}D}$ does not touch the ground (if we leave ${\displaystyle {}B,C}$ in their positions and put ${\displaystyle {}D}$ down to the ground, then ${\displaystyle {}A}$ would sink into the ground). We claim that we can bring the table, by turning it around its middle axis, into a position such that it stands on all four legs (we do not claim that the table is then horizontal). To see this, we consider the function which assigns to the angle of rotation, the height of ${\displaystyle {}D}$ above the ground, when the three other legs are (or would be) on the ground. This height might be negative (if we are on sand, this can be realized, else think of this "ideally“). In degree ${\displaystyle {}0}$, the height is positive. In degree ${\displaystyle {}90}$, we get a situation which is symmetric to the initial position, but still the legs ${\displaystyle {}A,B,C}$ are supposed to be on the ground. Hence, the height of ${\displaystyle {}D}$ is now negative. The function has, on the interval ${\displaystyle {}[0,90]}$, positive and also negative values. Since the function is continuous, by the Intermediate value theorem, it also has a zero.