My professor gave this word problem to us and I can't seem to navigate through all of the directions and hints to understand how to do it. I do know for sure he expects us to use the Intermediate Value Theorem.
Prove that at any instant of time, there are two antipodal points (two
points whose midpoint is the center of the earth) on the equator with the
same temperature, assuming that temperature changes continuously with
the points on earth.
Picture the equator as a unit circle and set up a coordinate system. Consider
a point P on the circle. We can represent the point P by the angle theta
between the line connecting the origin to the point P and the positive x-axis.
This way, the points on the equator may be labeled by angles theta in [0, 2pi].