Is there on the aquator two different places with the same temperature? [duplicate]

Question

I found this problem: "We consider the temperature on the equator as a continuous function of longitude. As mathematicians, we give longitudes in values ​​from 0 to 2π (instead of from 0◦ to 360◦) so the temperature is a continuous function T : [0, 2π] → R with T(0) = T(2π). Although it is known to be hot everywhere at the equator,we assume that it is not equally hot all around. Is there on the aquator two different places with the same temperature? Prove your statement."

I don't know if it is the length or what, but to me it is very confusing on how to even start to answer this, if anyone has an any Idea, please share!

Answer 1

We can write the temperature on the equator with a function $$f:\mathbb{R}\to\mathbb{R}$$and since f is obviously $2\pi$-periodic, we can write $$f(x)=f(x+2\pi)$$ Now let $$g:\mathbb{R}\to\mathbb{R},\\g(x)=f(x-\pi)-f(x)$$Note that if $g(x)=0$, the statement is proven. Now split this in 3 cases, depending on the values of $g(0)$, namely $g(0)<0$ or $g(0)=0$ or $g(o)>0$.

Then you just have to show that g has a zero in every case.