The Monster That Expands Our Mathematical Imaginations
On a recent episode of our podcast My Favorite Theorem, my cohost Kevin Knudson and I had the opportunity to talk with Ben Orlin, a math educator and author of the popular blog Math With Bad Drawings as well as two books, Math With Bad Drawings and Change Is the Only Constant. You can listen to the episode here or at kpknudson.com, where there is also a transcript.
Orlin decided to talk not about a theorem but about a favorite mathematical object, Weierstrass’s function. This function, sometimes known as a “monster,” answers the question of how closely continuity and differentiability are related. In mathematics, continuity is roughly what you might think it should be: a function is continuous if nearby inputs are sent to nearby outputs. (Is there a more rigorous definition? Yes! Here, if you insist.) A function is differentiable if at every point, you can find a tangent line, a straight line that approximates the function’s path near that point.
In rough terms, when you think about graphs of functions, a continuous function is one that doesn’t have jumps, and a differentiable function is one that doesn’t have corners or spikes. It seems clear that a function must be continuous in order to be differentiable A function with one corner in it—an example is the absolute value function f(x)=|x|, where |x|=x if x is greater than or equal to 0 and |x|= −x if x is less than 0—is continuous everywhere and differentiable everywhere except at x=0, where it has that corner.
It’s not too hard to cook up a function that has a lot of corners like that. You can make a sawtooth function with a peak or valley at every integer, for example. That function would be differentiable everywhere except at those isolated points, which are infinite in number but politely spaced out. Weierstrass wanted to know whether there was a limit to how not differentiable a continuous function could be, and this example shows that it can be pretty darn non-differentiable! While the function is continuous everywhere, it is not differentiable at any point.
To be pedantic, it is not quite accurate to say the Weierstrass function. Weierstrass’s original construction allowed for two parameters to be chosen, so there is a whole family of these functions. Since Weierstrass first published his curves, other mathematicians have defined more such monsters, and even proved that in a sense, most continuous curves are nowhere-differentiable. It’s a blow to those of us who like our math neat and tidy, but perhaps we can think of it as an invitation to think bigger and weirder about what we should expect in mathematics.
In each episode of My Favorite Theorem, we ask our guest to pair their theorem with something. You’ll have to check out the episode to see why Orlin thinks molecular gastronomy is the ideal accompaniment to Weierstrass’s function.