Do infinitely many points on earth have the same temperature as their antipodal?

Take a point $p in X setminus A$ and call it northpole, call $-p$ southpole. Now we look at the longitudes, the circles (longitudes) through these poles on the sphere.

Take one longitude and call it $L$.

$f$ is defined on $L$ and it is nonzero on $p$. As $f(-p) = -f(p)$ we know that $f$ has both negative and positive points on $L$. As $L$ is connected and $f$ is continous this means that there is a point $z_L$ on $L$ where $f (z_L)=0$. So $ z_L in L cap A$. $p$ and $-p$ are not in $L cap A$, though, so $z_L$ is not one of the poles.

As this is true for each of the infinite longitudes $L$ this means that there is a point in $A$ for every longitude. These points are distinct as the longitudes only meet in the poles (which are not in $A$). Therefore $A$ must have at least as many points as there are longitudes, so $A$ is inifinite.

qed

(Slightly away from your specific question, if you are working on something as similar to the Borsuk-Ulam theorem then you can't do yourself a bigger favour then watching this most excelent video from 3b1b about it: https://www.youtube.com/watch?v=yuVqxCSsE7c)