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. 

Hint: consider the function 
$$
\begin{align}
f:E&\to \mathbb R\\
x&\mapsto T(x)-T(-x)
\end{align}
$$
Now, trying the intermediate value theorem at the correct points should give you a proof.
