Let the difference of temperature between angle θ and angle θ+π be f(θ). If f(θ)=0 when θ=0, we have already found two antipodal points which have a same temperature. If f(0) is not 0, then f(0)>0 f(π)<0 or f(0)<0 f(π)>0. The circle is a closed figure, so the graph which show the relationship between θ and f(θ) is continuous, no matter what the graph looks like. The change of θ is similar with a running race. When you are the NO.10 initially and your aim is NO.1, you need to exceed everyone in front of you. The first step is to exceed the one who is right in front of you and become NO.9. Repeat this process several times, you will be the NO.1. When we review the whole process, you have at least once to be in each position on the court, because you keep running during the race. As same as the graph, when you start at f(0) and try to reach f(π), you have to experience each point between f(0) and f(π) at least once. As we known, f(0) and f(π) have different sign, so 0 must between them. That is why there exist a number a in (0,π) such that f(a)=0.

A Function:
Bob has worked for A company for 2 years. He got $100000 for his salary. Now his employer decides to raise his salary because of his ability. After arising, company pays him $5000 per month.

It is the function between the time(t) and money (f) he owns. f(t)=100000+5000t

A sequence:
A biology student is observing cell division. There is only one bacterium cell in a petri dish. Each second, one cell division occurs. After first division, two bacterium cells appear in the dish.
The sequence is n=1*2^(t-1)

A series:
A boss invests a really effective and risky task which monthly benefit is 1%. He spends $100,000,000 to invest initially. Because of the risk, after each month he takes 20% out of his investment(not include the benefit) to minimize the risk.

There is a series which is between time(t) and benefit(y).

Does the graph of your function have a horizontal asymptote?
No, because it is a linear function.
Does your sequence converge?
No, since it is a geometric sequence which has a common ratio of 2 being greater than 1.
Does your series converge?
Yes, it is a geometric series and the absolute value of r is smaller than 1.