Today, you too get to learn what my real analysis students are doing in class. The intermediate value theorem is a result in pure mathematics that is extremely intuitive.

If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u.

See, that’s the most intuitive thing you’ve ever seen, right? Let me try to explain it with a few more words. Say we have some quantity which we can measure – say temperature – which varies continuously (no sudden changes) as we change another quantity – say time – then if the temperature is 5 degrees at midnight (in whatever you feel like measuring temperature in) and 25 degrees at midday (if you’re using Fahrenheit, you deserve to be that cold) then for any temperature between 5 and 25 degrees, there is a time between midnight and midday that was this temperature. Makes sense, right? If it gets continually warmer from midnight until midday, then it has to hit every temperature in between.

Let’s consider sound as another example. If the sound pressure level at a rock gig is 125dB in the front row and 60db on a neighbours balcony – who for some reason insists on complaining about the noise to authorities despite choosing to move in next to a rock venue – then there must be some point in between where the sound will be at a volume of 85dB (or any other arbitrarily chosen number between 60 and 125).

A good exercise for undergrads, is to use this theorem to prove that there exists two opposite points (antipodes) on the equator with the exact same temperature. To demonstrate this, consider the temperature difference between any point on the equator and its antipode. Now this difference could be either positive or negative depending on whether or not the antipode is warmer or cooler than the point we’re looking at. It could also be zero, but if the temperature difference is zero then we’ve amazingly chosen two opposite points with the same temperature and there’s nothing to prove! We know this temperature difference (between a point and its antipode) should vary continuously as we move around the equator. At some point the difference must be positive, that is, the point we’re looking at is warmer than it’s antipode. Now let’s consider the difference between another point and its own antipode. If we choose the new point to be the original point’s antipode, then we know the antipode of this point (the original point) is warmer, so the difference must be negative. So now the difference between two opposite points is positive somewhere on the equator, and negative somewhere else on the equator; this means that starting from somewhere in between, the difference must be zero. So we’ve established there must be two opposite points with the same temperature.

Congratulations, you now know a theorem in pure mathematics!

Interestingly, this relates to the fact that a table can be made stable regardless of how wobbly the ground is. There’s a pretty cool mathematical paper explaining this here – for the mathematically courageous.