At any time, there exist two points on the surface of the Earth, which have
exactly the same temperature.

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, therefore there exists some temperature T such that T(a)<T<T(b).
Without loss of generality we can pick T such that T>T(a) and T>T(a'). By the
IVT, 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 we've seen that there exist two points, how about infinitely many points?
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
isothermal curve.