At every given point in time there are two points on the equator of the Earth that have the same temperature.

How do we know this? Well, here's a proof. Let's look at the equatorial plane which slices through the Earth at the equator. The equator is a circle which lies in that plane, and we can choose a coordinate system on the plane so that the point $(0,0)$ lies at the centre of the equator. For each point $x$ on the equatorial circle there is a point $-x$ which lies diametrically opposite $x$. Now each point $x$ on the equator comes with a temperature $t(x)$. We can assume that the function $t$, which allocates a temperature to each point, is continuous. That's because temperature doesn't suddenly jump up or down as you move around on the Earth. Now consider the function $$f(x) = t(x)-t(-x).$$ It is also continuous. If this function is equal to $0$ for some point $x$, then we are done because if $$f(x) = t(x)-t(-x)=0$$ then $$t(x)= t(-x),$$ so the temperature at $x$ is the same as the temperature at $-x$. If $f(x)$ isn't equal to $0$ anywhere, then let's assume (without loss of generality) that there is a point $x$ at which $f(x)>0,$ so $$f(x) = t(x)-t(-x)>0.$$ This implies that $$f(-x) = t(-x)-t(x)=-f(x)

Illustration of the intermediate value theorem. If t(x)>0 and t(y) and t is continuous, then there is a point z between x and y such that t(z)=0.

Thus, since $f(-x)0$, there must be a point $y$ on the circle such that $f(y)=0$. So $$f(y) = t(y)-t(-y)=0$$ which means that $$t(y)=t(-y).$$ So the temperature at the point $y$ is the same as the temperature at the point $-y.$

The result actually holds for any circle on the Earth, not just the equator. In fact, the result is the one-dimensional case of the Borsuk-Ulam theorem, which says that for any continuous function $t$ from the circle to the real numbers there is a point $x$ such that $t(x)=t(-x).$

The more general version of the Borsuk-Ulam theorem says that for any continuous function $t$ from the $n$-sphere to the set of $n$-tuples of real numbers there is a point $x$ such that $t(x)=t(-x)$.