I’ve teased this one before.
So you know how the Earth is a sphere, but from our normal vantage point right up close to its surface it looks flat? That happens with functions too. Here I mean the normal kinds of functions we deal with, ones with domains that are the real numbers or a Euclidean space. And ranges that are real numbers. The functions you can draw on a sheet of paper with some wiggly bits. Let the function wiggle as much as you want. Pick a part of it and zoom in close. That zoomed-in part will look straight. If it doesn’t look straight, zoom in closer.
We rely on this. Functions that are straight, or at least straight enough, are easy to work with. We can do calculus on them. We can do analysis on them. Functions with plots that look like straight lines are easy to work with. Often the best approach to working with the function you’re interested in is to approximate it with an easy-to-work-with function. I bet it’ll be a polynomial. That serves us well. Polynomials are these continuous functions. They’re differentiable. They’re smooth.
That thing about the Earth looking flat, though? That’s a lie. I’ve never been to any of the really great cuts in the Earth’s surface, but I have been to some decent gorges. I went to grad school in the Hudson River Valley. I’ve driven I-80 over Pennsylvania’s scariest bridges. There’s points where the surface of the Earth just drops a great distance between your one footstep and your last.
Functions do that too. We can have points where a function isn’t differentiable, where it’s impossible to define the direction it’s headed. We can have points where a function isn’t continuous, where it jumps from one region of values to another region. Everyone knows this. We can’t dismiss those as abberations not worthy of the name “function”; too many of them are too useful. Typically we handle this by admitting there’s points that aren’t continuous and we chop the function up. We make it into a couple of functions, each stretching from discontinuity to discontinuity. Between them we have continuous region and we can go about our business as before.
Then came the 19th century when things got crazy. This particular craziness we credit to Karl Weierstrass. Weierstrass’s name is all over 19th century analysis. He had that talent for probing the limits of our intuition about basic mathematical ideas. We have a calculus that is logically rigorous because he found great counterexamples to what we had assumed without proving.
The Weierstrass function challenges this idea that any function is going to eventually level out. Or that we can even smooth a function out into basically straight, predictable chunks in-between sudden changes of direction. The function is continuous everywhere; you can draw it perfectly without lifting your pen from paper. But it always looks like a zig-zag pattern, jumping around like it was always randomly deciding whether to go up or down next. Zoom in on any patch and it still jumps around, zig-zagging up and down. There’s never an interval where it’s always moving up, or always moving down, or even just staying constant.
Despite being continuous it’s not differentiable. I’ve described that casually as it being impossible to predict where the function is going. That’s an abuse of words, yes. The function is defined. Its value at a point isn’t any more random than the value of “x2” is for any particular x. The unpredictability I’m talking about here is a side effect of ignorance. Imagine I showed you a plot of “x2” with a part of it concealed and asked you to fill in the gap. You’d probably do pretty well estimating it. The Weierstrass function, though? No; your guess would be lousy. My guess would be lousy too.
That’s a weird thing to have happen. A century and a half later it’s still weird. It gets weirder. The Weierstrass function isn’t differentiable generally. But there are exceptions. There are little dots of differentiability, where the rate at which the function changes is known. Not intervals, though. Single points. This is crazy. Derivatives are about how a function changes. We work out what they should even mean by thinking of a function’s value on strips of the domain. Those strips are small, but they’re still, you know, strips. But on almost all of that strip the derivative isn’t defined. It’s only at isolated points, a set with measure zero, that this derivative even exists. It evokes the medieval Mysteries, of how we are supposed to try, even though we know we shall fail, to understand how God can have contradictory properties.
It’s not quite that Mysterious here. Properties like this challenge our intuition, if we’ve gotten any. Once we’ve laid out good definitions for ideas like “derivative” and “continuous” and “limit” and “function” we can work out whether results like this make sense. And they — well, they follow. We can avoid weird conclusions like this, but at the cost of messing up our definitions for what a “function” and other things are. Making those useless. For the mathematical world to make sense, we have to change our idea of what quite makes sense.
That’s all right. When we look close we realize the Earth around us is never flat. Even reasonably flat areas have slight rises and falls. The ends of properties are marked with curbs or ditches, and bordered by streets that rise to a center. Look closely even at the dirt and we notice that as level as it gets there are still rocks and scratches in the ground, clumps of dirt an infinitesimal bit higher here and lower there. The flatness of the Earth around us is a useful tool, but we miss a lot by pretending it’s everything. The Weierstrass function is one of the ways a student mathematician learns that while smooth, predictable functions are essential, there is much more out there.