TITLE: Intermediate Value Theorem question
QUESTION [5 upvotes]: The temperature $T(x)$ at each point $x$ on the surface of Mars (a sphere) is a continuous function. Show that there is a point $x$ on the surface such that $T(x)=T(-x)$
(Hint: Represent the surface of Mars as $\{x\in{\mathbb{R}^3}:||x||=1\}$.)
Consider the function $f(x)=T(x)-T(-x)$

So.....
I consider an unit sphere is locating at the origin of a $xyz$-plane. 
As $||x||=1$, I can say with $radius = 1 = \sqrt{x^2+y^2+z^2}$
To find there is a point $T(x)=T(-x)$, we use the formula $f(x)=T(x)-T(-x)$ and show somehow $f(x)$ will equal to $0$ ???
It will be a point in the upper hemisphere and another point with the exactly opposite vector (if using $ijk$ plane) on the lower hemisphere

REPLY [3 votes]: Hint: Pick some point $x$. If you are lucky, $f(x) = 0$, but probably this does not occur. In the other case, what can you say about the relation between $f(x)$ and $f(-x)$? Now connect $x$ and $-x$ using a path along the sphere...