My Favorite One-Liners: Part 72

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

In calculus, the Intermediate Value Theorem states that if f is a continuous function on the closed interval [a,b] and y_0 is any number between f(a) and f(b), then there is at least one point c \in [a,b] so that $f(c) =y_0$.

When I first teach this, I’ll draw some kind of crude diagram on the board:

In this picture, f(a) is less than y_0 while f(b) is greater than y_0. Hence the one-liner:

I call the Intermediate Value Theorem the Goldilocks principle. After all, f(a) is too low, and f(b) is too high, but there is some point in between that is just right.




A Natural Function with Discontinuities: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on a natural function that nevertheless has discontinuities.

Part 1: Introduction

Part 2: Derivation of this piecewise function, beginning.

Part 3: Derivation of the piecewise function, ending.




A natural function with discontinuities (Part 1)

The following tidbit that was published on the American Mathematical Monthly’s Facebook page caught my attention:


Here’s the relationship between r, R, and \theta in case it isn’t clear from the description. The gray sector is determined by r and \theta, and then the blue circle with radius r is chosen to enclose the sector.


Unfortunately, there was typo for the third case; it should have been r = R \sin \frac{1}{2} \theta if 90^\circ \le \theta \le 180^\circ. Here’s the graph if R = 1, using radians instead of degrees:


As indicated in the article, there’s a discontinuity at t=0. However, the rest of the graph looks nice and smooth.

Here’s the graph of the first derivative:


The first derivative is continuous (and so the original graph is smooth). However, there are obvious corners in the graph of the first derivative, which betray discontinuities in the graph of the second derivative: