there exist two antipodal points on the equator that have the same temperature.

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

Let $\theta$ denote longitude; it varies from $+180$ to $-180$. Define $f(\theta)$ to be the temperature at longitude $\theta$, minus the temperature at the antipodal point to longitude $\theta$. If ever $f(\theta)=0$, you are done. However, $f(\theta)=-f(\theta')$, where $\theta'$ is antipodal to $\theta$. So, continuously vary $\theta$ to get $\theta'$, and apply IVT(Intermediate Value Theorem).