I've been thinking about this for awhile now (I am trying to find a method of proving the Borsuk-Ulam theorem in 2 dimensions without resorting to the usual, and not so intuitive to non-mathematicians topological arguments). I saw the question Does a continuous scalar field on a sphere have continuous loop of "isothermic antipodes" posted here, but am curious about a stronger statement.
Say you have a continuous map $f$ from $S^2$ to $R$. Let $U$ be the set of points such that $f(x)=f(-x)$ for $-x$ being the antipode of $x$. I;ve been able to show that if $U$ is not the entire sphere, then every pair of antipodal points which are not mapped to the same value by $f$ are disconnected in $S^2-U$ (using an argument similar to one posted in the above question). So; the set $U$ must then contain at least one continuous loop. What I am trying to show is that $U$ must contain a loop $C$ such that $C=-C$ (equivelantly, that there is a loop $C$ in $U$ for which $S^2-C$ is two disjoint "hemispheres" of equal areas).
It seems to me that if this were not the case then you could construct a second map $g$ on $S^2$ such that its set of antipodes mapping to the same value would be completely disjoint from $U$ (since two loops on a sphere which are not "equators" in the sense that they divide the area of the sphere in half do not necessarily need to intersect), and then you could construct a map from $S^2$ to $R^2$ which had no antipodes mapping to the same value by $F=(f,g)$, which would contradict the theorem I know is true but would like to prove.
Maybe I'm going about this in a difficult way, so any suggestions on how to study the nature of the set $U$ would be appreciated.