Sometimes I am amazed by the power and beauty of Math. Yet, the statement above is a true one!!
Proof:
Let T(x) be the temperature function for a point x on the Globe.
Consider a path p(t) from a "cool" point "a" on the left side of the Globe, to point "b", where T(b) is the maximum temp of the Globe. Consider also a path p'(t) from another "cool" point "a'" on the other side, to "b".
The function T(p(t)), with t describing the path p(t) is continuous and T(a)<T(b) by assumption. => There exists some temperature T such that T(a)<T<T(b). Without loss of generalization we can pick T such that T>T(a) and T>T(a').By the Intermediate Value Theorem, there exists a point "c", somewhere along the path p(t), such that T=T(c).
The same argument applies to the other half, if we consider it applied
to path p'(t) from a' to b, implying the existence of a point c', such
that T=T(c')=T(c).
QED.
Now that you've seen that there exist two points, how about infinitely many points? Easy. Repeat the argument above (using density), for all points a(t) and a'(t) on both sides of the globe, to get a simple closed curve j(t) of points which have the same temperature at any given moment!! This curve would be what physicists call "an isothermic curve".