Prove that on every great circle on the earth there are antipodal points at which the temperature is the same.

Question

Prove that on every great circle on the earth there are antipodal points at which the temperature is the same. Assume that the surface of the earth is a sphere and that the temperature is a continuous function.

---

Side note: This question is on an assignment focusing on connectedness and path-connectedness of sets in $\Bbb{R}^n$ but I'm not sure how this relates. Any insight would be greatly appreciated.

Answer 1

Let $\phi(x) = t(x)-t(-x)$, where $t$ is the temperature function.

Since $\phi$ is continuous and $\phi(x) = - \phi(x)$ we can use the intermediate value theorem to find some $y$ such that $\phi(y) = 0$.

Answer 2

Hint:

Hint 2:

Intermediate Value Theorem.

Answer 3

let f(x) = temperature at x - temperature at antipode of x.

f(antipode of x) = -f(x)

If we assume changes in temperatures are gradual with location, we can claim f is continuous.

if f(a) > 0 for some a then f(a at antipode) = -f(a) < 0

Intermediate value theorem says f(x) = 0 for some value of x.

Answer 4

Here's a sledgehammer you can hit it with, if you've already covered it in your course: the Borsuk-Ulam theorem lets you conclude exactly what you want. The theorem says that any continuous $S^n \to \mathbb{R}$ maps some pair of antipodal points to the same value.

For $n=1$ however, as other answers show, you don't need such heavy machinery, and the intermediate value theorem does the trick.