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.