First of all, let us restate the question. The question says that in any circle in the universe, we can find antipodal points on the circle of the same temperature. In order to solve this problem, let’s draw a function of it. Which f(x)=T(θ)-T(θ+π), which the T(θ) is any point on the circle and T(θ+π) is the antipodal point of the specific point. In this case, if the function f(x) equals 0, it means T(θ) equals T(θ+π). However, we can’t directly prove the function equals 0. Therefore, we need to use another way to prove it.

It would better if we draw a picture first.

Then, let (0, infinity) of y axis be the positive temperature and  (-infinity, 0) of y axis be the negative temperature. The point T(0) starts moving in the positive interval of the y axis, so that we always can get a positive temperature. Oppositely, we will get a negative temperature T(π). Therefore, we can clearly guarantee that f(0)>0 because a positive number minus a negative number always get a positive number. Similarly, if we move T(π) in the negative interval, T(2π) will move in the positive interval, which f(π)<0.

Before we start to use a theorem to prove it, let me introduce it to you. If we have a function continuous in an interval, we can guarantee that there must have a number x among the interval that corresponds to a number L. This concept may be hard for you to understand. Let me give you an example. If you want to fly from the southern hemisphere to the northern hemisphere, you must have to cross the equator no matter how the pilot drives the airplane. In this case, if we can prove that f(0) is opposite with f(π), we can prove that there must have a value of x=a corresponding to f(a)=0. Therefore, we have already proved that f(0) and f(π) are the opposite sign, we just need to guarantee f(x) is continuous. If we consider the function, then there would always have a temperature on the circle, and the temperature increases or decreases constantly. Therefore, we can directly guarantee that f(x) is continuous so that we can conclude the temperature at Pa is equal to the temperature at Pa+π.