I was jaywalking with an old math professor of mine across a busy road in Athens, GA recently (aside, ask me about my favorite things about Athens. I’ll talk your ear off.) It struck me that despite the (slightly) dangerous situation, I could find solace in the Intermediate Value Theorem (IVT) to keep me safe.
If you don’t remember the IVT from high school calculus, it goes as follows: if \(f\) is a continuous function in the domain \([a, b]\), it must take on any given value \(f(x)\) between \(f(a)\) and \(f(b)\) for some \(x \in [a, b]\). For example, if a continuous function runs through the domain \([1, 2]\) where \(f(1) = 0\) and \(f(2) = 100\) then there must be some \(x \in [0, 1]\) for which \(f(x) = 5\) or 15 or pi or any other value between 0 and 100 (you can convince yourself of this pretty quickly by drawing out a continuous function on a piece of paper).
But the IVT need not be applied only to the completely contrived and otherwise uselessly inapplicable field of calculus (sorry, Newton). It also has the immensely useful application of assuaging your jay-walking fears. Roads are really nothing more than (generally) continuous, (generally) smooth functions – the domains upon which cars (generally) drive (and hardened jaywalkers walk). To get to my position along the domain (call this \(s\)) from some car’s position (call this \(c\)), I know that the car must pass through all points between \(f(c)\) and \(f(s)\). In other words, as long as I correctly understand \(f(c)\), \(f(s)\), and have an alright idea of the car’s velocity and acceleration (the first and second derivatives of \(f(c)\) respectively), I should have plenty of time to decide when to jaywalk and how quickly to do so.
But Sudhan, I hear you saying, this is trivial! I could’ve told you this without thinking about continuity and the IVT. And indeed, you could have. But this is a testament to how much we take the IVT for granted! For a second, imagine a world in which the IVT wasn’t a given. Jaywalking (and traveling in general) would have infinitely greater risks. Think for a moment about the effects this might have beyond transportation. The way we see things is little more than rays of light hitting our eyes after bouncing around all sorts of objects in our environment. What if those rays of light didn’t have to travel through the entirety of the domain between their origin and our eyes before they entered our vision. We would be stochastically blind!
The IVT may not be the biggest, baddest, most complicated theorem on the block. But that doesn’t mean it’s not beautiful and wonderful in all its own ways. If you need convincing of this, step out into traffic after reading this blog post (for legal reasons, don’t actually do this) and see how much better you feel knowing that the IVT is keeping you safe.