beavers Who are Busy

Many of you will be familiar with the famous counterexample to the claim that all differentiable functions are continuously so: f(x) = (x^2) sin(1/x), with the point (0,0)

When you look for ideas on how to show that this is differentiable at 0, despite the fact that the derivatives do not approach the value of this derivative when you get closer to 0, you often are met with the evaluation of the derivative at all points other than 0 as well as at 0.

I think this doesn’t give a really full picture of what’s going on, so I’d like to share with you a way of approximating the behavior of this function with a set of successively compressed trapezoids.

The basic idea is this: place the repeated segment of the sinusoidal curve inside of a trapezoid, and squish and stretch that trapezoid to match the parabola it’s near, and use the asymptotic behavior of the dimensions of these trapezoids to understand the behavior of the derivatives as x gets closer to 0.

To understand where to place and use the trapezoids, we need to see where the function x^2*sin(1/x) is tangent to the parabola x^2. This is simple enough, and we see that this happens every x = (2/pi)(1/(4k+1)), where k is an integer.

This tells us that we are working with trapezoids of dimension (2/pi)((1/4k+1)-(1/4k+5)) in width and (8/pi^2)(1/4k+1)^2 in “height”.

Asymptotically, these trapezoids tend towards similarity to a 1 by pi rectangle, the exact same dimensions as sin(x) normally has!

Basically, this tells us that, as they approach 0, the sinusoidal curves retain their full range of motion, so we know that the derivative wobbles between 1 and -1 infinitesimally quickly as the function approaches 0, despite the fact that, at 0, the derivative is 0.

I thought that this problem was a great example of how there are often ways of intuitively understanding facts from real analysis with restatements in the language of normal geometry.