antipode temperature equation Problem: 
For any point $x$ on the sphere, the antipode, $-x$, is the point which is exactly on the other side. 
$E =$ surface of the sphere. 
Let $T : E \to R$ be a continuous function where $T(x)$ represents the temperature at the point $x.$ Prove that there exists a point $x$ on the equator such that $T(x) = T(-x).$
My idea: Maybe we can assume there doesn't exist such point $x$ and show that this results in that $T$ is not a continuous function anymore, which is a contradiction!
I'm not sure how to show this though. 

Consider the result that a continuous bijection between a compact space and a Hausdorff space is a homeomorphism. Now, you can consider a map T:$S^2\rightarrow \mathbb R$ as a map from $S^2$ onto $T(S^2)$. If this map were an injection, then it would be an embedding of the compact space $S^2$ into the Hausdorff space $\mathbb R$ (or, more accurately, the (Hausdorff) subspace $T(S^2)$ , which is not possible for many reasons --not even $S^1$ can be embedded into $\mathbb R$
See:http://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem
