The Monster That Expands Our Mathematical Imaginations

On a new episode of our podcast My Favourite Theorem, my cohost Kevin Knudson and I experienced the possibility to speak with Ben Orlin, a math educator and creator of the popular blog Math With Lousy Drawings as well as two publications, Math With Lousy Drawings and Change Is the Only Consistent. You can listen to the episode here or at kpknudson.com, where there is also a transcript.

Orlin decided to speak not about a theorem but about a most loved mathematical object, Weierstrass’s function. This function, from time to time identified as a “monster,” solutions the query of how closely continuity and differentiability are related. In arithmetic, continuity is about what you may well feel it should be: a function is continual if close by inputs are despatched to close by outputs. (Is there a more demanding definition? Of course! Here, if you insist.) A function is differentiable if at each issue, you can obtain a tangent line, a straight line that approximates the function’s route around that issue.

In rough terms, when you feel about graphs of functions, a continual function is one that does not have jumps, and a differentiable function is one that does not have corners or spikes. It looks crystal clear that a function have to be continual in purchase to be differentiable A function with one corner in it—an example is the absolute worth function f(x)=|x|, where |x|=x if x is better than or equal to and |x|= −x if x is fewer than 0—is continual all over the place and differentiable all over the place besides at x=, where it has that corner.

It’s not too really hard to prepare dinner up a function that has a large amount of corners like that. You can make a sawtooth function with a peak or valley at each integer, for illustration. That function would be differentiable all over the place besides at these isolated points, which are infinite in range but politely spaced out. Weierstrass required to know whether or not there was a restrict to how not differentiable a continual function could be, and this illustration exhibits that it can be rather darn non-differentiable! While the function is continual all over the place, it is not differentiable at any issue.

An illustration of the Weierstrass function, displaying the way its cragginess exhibits up at each scale. Credit score: Eeyore22 Wikimedia

To be pedantic, it is not pretty exact to say the Weierstrass function. Weierstrass’s authentic design permitted for two parameters to be picked out, so there is a full family members of these functions. Due to the fact Weierstrass first printed his curves, other mathematicians have described more this kind of monsters, and even proved that in a sense, most continuous curves are nowhere-differentiable. It’s a blow to these of us who like our math neat and tidy, but perhaps we can feel of it as an invitation to feel bigger and weirder about what we should expect in arithmetic.

In each and every episode of My Favourite Theorem, we inquire our visitor to pair their theorem with one thing. You will have to test out the episode to see why Orlin thinks molecular gastronomy is the perfect accompaniment to Weierstrass’s function.