TITLE: there exist two antipodal points on the equator that have the same temperature.
QUESTION [7 upvotes]: Argue that there exist at any time two antipodal points on the equator that have the same temperature. 
The temperature function can be assumed to be continuous.
I am supposed to use the mean value theorem, but I don't see how. Thanks in advance.

REPLY [16 votes]: Okay, so lets just consider points along the equator, and  let $  t:\left [ 0,2\pi  \right ]\rightarrow \mathbb{R}$ be the temperature at a point with angle $\theta$ from some predetermined point on the equator.
Now, we are given that t is continuous in $\theta$ on $\left [ 0,2\pi  \right ]$, and we see that t is $2\pi$ periodic.
Define $T:\left [ 0,2\pi  \right ]\rightarrow \mathbb{R}$ to be the antipodal difference in temperature, that is $T := t(\theta + \pi) - t(\theta)$
Then T is also continuous on $\left [ 0,2\pi  \right ]$, and we have that:
$T(0) = t(\pi) - t(0)$ and $T(\pi) = t(2\pi) - t(\pi)$ 
So as t is $2\pi$ periodic, we get that $T(0) = -T(\pi)$
If $T(0) =  0$ then we are done and we have our antipodal points with equal temperature, otherwise if $T(0) \neq 0$, then as $T$ is continuous on $\left [ 0,\pi  \right ] \subset$ $\left [ 0,2\pi  \right ]$ and without loss of generality  $T(0) < 0 < T(\pi)$, then $\exists \alpha \in \left [ 0,\pi  \right ]$ such that $T(\alpha) = 0$.
And then $t(\alpha) = t(\alpha + \pi)$ so we have found our antipodal points with the same temperature.
This is really just a specific case of a really interesting theorem called the Borsuk–Ulam Theorem, which makes similar sorts of statements for n-dimensional spheres mapping to n-dimensional planes. Here we have a 2 dimensional sphere mapping to a 1 dimensional plane, but we considered a 1 dimensional subsphere (our equator), and the Borsuk–Ulam Theorem says on any continuous mapping of an n-dimensional sphere to an n-dimensional plane, there will be two antipodal points who get mapped to the same point.
http://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem