Intermediate Value Theorem

(You can watch the video version of this post on YouTube.)

Now here’s a theorem that’s fairly obvious? Especially if you don’t try to generalise it, but like, as a tactical move it’s useful to be aware of. As the shape of an argument.

So let $f$ be real function that’s continuous on the interval $[a,b]$. Then on this interval it must take ALL values between $f(a)$ and $f(b)$. It’s often used existentially, saying that a particularly function value that we like must be attained. It’s called the Intermediate Value Theorem. But this is obvious, right? The function is continuous, just look at the picture. Well, yeah, so if you can reduce your problem .. to THIS, you’re done.

Let’s keep it simple, say you’re given a bounded object in 2D. Then for ANY direction, there exists a line that cuts the object precisely in half by area. Proof: Sweep the line over the plane and consider the function of how much of the object’s area you’ve swept over. This is a continuous function that goes from zero to a 100%, so it must be 50% somewhere.

OK, maybe that was a bit lame, but this cutting thing can be generalised to n objects simultaneously in $n$ dimensions and this gets you the fabulously-named Ham Sandwich theorem - uh, look it up sometime - but let’s end this video instead with a different application. I had heard of this before, but had never realised it’s quite this easy to prove.

Theorem: Let $f$ be a continuous function from the sphere to $\mathbb{R}$. Then there exists a pair of antipodal points - points opposite each other on the sphere - that have the same function value.

Proof. Take an arbitrary pair of antipodal points $p$ and $q$. If $f(p)=f(q)$, we are done, otherwise let $d$ be the difference. Now pick a great circle through both points and rotate them along this circle. After half a turn they have swapped places, so the sign of $d$ is flipped. Then by the intermediate value theorem, somewhere along this rotation, $d$ was zero. QED: these are your points.

You can even to this for two functions from the sphere to R and if you generalise that even further you get the Borsuk-Ulam theorem, but I’ll let you look that up on your own.