Weird Functions; why continuity matters so much

This summer I have been working through a delightful book by Gelbaum and Olmsted called Counterexamples in Analysis. No, I don't own stock in the Dover book company, nor do I know either of the authors.

I can recommend this book to anyone who is either learning elementary analysis, interested in learning the "whys" behind calculus, or is teaching calculus, real analysis or complex variables.

I'll share a few cool examples that I was not familiar with prior to reading this book.

• A function that is continuous precisely at the irrational numbers whose discontinuities are removable.

A reminder: if the limit as "x" approaches "a" of a function "f" exists but is not equal to the value of the function at "a", we say that the discontinuity is "removable". For example, the function f(x) = (x^2 -1)/(x + 1) has a removable discontinuity at x = 1 because f(1) doesn't exist (zero in the denominator) but the limit of f as x approaches 1 is clearly zero.

When we bring this concept up in calculus, we show a graph with a single "hole" where (a, f(a)) should be.

Now recall that a rational number is a number that can be put in the form p/q where p and q are integers and p, q have no common factors other than 1 or -1. If we insist that, say, q be positive, then this form is unique.

Now define f(x) to be "1/q" if x = p/q (i.e., x is rational) and let f(x) = 0 if x is irrational (e. g. f(pi) = 0).

Then note that f is continuous at every irrational number; it is a wonderful delta-epsilon argument to see this. Also note that if r is any rational number and we redefine f(r)=0 and leave f alone elsewhere, f becomes continuous at r. Hence each discontinuity is removable.

• A function that is bounded at every point but is unbounded on any interval (closed or open; we don't allow for empty or single point intervals)

Now let g(x) = q if x is rational and x = p/q and let g(x) = 0 if x is irrational.

Note that any interval whatsoever contains rational numbers of arbitrarily high magnitude (to see this, divide the real number line into, say, 1/10'ths, 1/100'ths, 1/1000's and so on, the way you would a ruler). Hence g is unbounded on any given interval, even though g(x) is finite for any given x.

Cameron Gordon

Photo: Cameron Gordon (my Ph. D. advisor) with about half of his students at a conference for his 60'th birthday. I am the one with long hair, a gray beard, and a black t-shirt with planets on it.

Cameron Gordon is one of knot theory's top all time researchers. He is probably best known the Gordon-Lueke result (John Lueke was one of his top students and is the blonde guy behind me and to the left) which states that two knots are equivalent if and only if their complements are homeomorphic.

To the non-specialist: a knot will be considered as a smooth (smooth as meaning "differentiable") closed loop in the three sphere (which we consider as the standard 3 space with "infinity" considered to be a point; to visualize this, go one dimension down. Look at the plane and imagine what would happen if "infinity" was declared to be a point. That point could be, say, the north pole). Because of the differentiability condition, one can think of a knot as something like a shoestring whose ends have been fused.

A knot complement is what is left when the shoestring is removed from three space. So the Gordon-Lueke result says that if two knot complements have a one to one continuous function between them that has a continuous inverse, then the knots themselves must have been the "same knot".

Note: this result is well known to be FALSE for "links" (systems of more than one closed loop).

Hyperbolic Space and Crochet?

From the New York Times:

Professor Lets Her Fingers Do the Talking

By MICHELLE YORKITHACA, N.Y. - Some people looking at the crocheted objects on Daina Taimina's kitchen table would see funky modern art. Others would see advanced geometry.
The curvy creations, made of yarn, are actually both. And they are helping two very different groups - artists and mathematicians - learn more about each other. Increasingly, they are also making a quirky celebrity out of the woman who created them.

"The forms are amazing," said Binnie B. Fry, the gallery director of the Eleven Eleven Sculpture Space, an art gallery in Washington, where Dr. Taimina's creations are part of a summer exhibition called "Not the Knitting You Know."

Dr. Taimina, a math researcher at Cornell University, started crocheting the objects so her students could visualize something called hyperbolic space, which is an advanced geometric shape with constant negative curvature. Say what?

Well, balls and oranges, for example, have constant positive curvature. A flat table has zero curvature. And some things, like ruffled lettuce leaves, sea slugs and cancer cells, have negative curvatures.

This is not some abstract - or frightening - math lesson. Hyperbolic space is useful to many professionals - engineers, architects and landscapers, among others. So Dr. Taimina expected some attention for her yarn work, especially from math students destined for those professions. But her work has recently drawn interest from crocheting enthusiasts.

Math professors have been teaching about hyperbolic space for decades, but did not think it was possible to create an exact physical model. In the 1970's, some educators, including Dr. Taimina's husband, David Henderson, a math professor at Cornell, created hyperbolic models, but the first ones, made from paper and cellophane tape, were too fragile to be of much use.
Though she did not realize it at the time, Dr. Taimina was a good candidate to create a better model. As a precocious child in her native Latvia, she tried her elementary school teacher's patience. When her fellow second graders did not understand a math lesson, she recalled, she would jump up and yell, "I can't stand these idiots," prompting her teacher to send notes home.
By high school she had settled down, and was most impressed by a teacher who was known to keep his advanced students laughing and engaged. When she became an educator, she decided that no student, regardless of aptitude level, would feel out of place in her classroom. One way she assured that was by using everyday objects to explain theories. (She was known for peering so intently at the oranges at her local grocery to see if she could find perfectly round ones to use in her geometry class that she scared the clerks.)

But it was her crocheting hobby that would prove really useful and make her something of a star - at least to knitters and math lovers.

In 1997, while on a camping trip with her husband, she started crocheting a simple chain, believing that it might yield a hyperbolic model that could be handled without losing its original shape. She added stitches in a precise formula, keeping the yarn tight and the stitches small. After many flicks of her crocheting needle, out came a model.

One professor who had taught hyperbolic space for years saw one and said, "Oh, so that's how they look," Dr. Taimina recalled in an interview at her home here, not far from the Cornell campus. A year after she created the models, she and her husband gave a talk about them to mathematicians at a workshop at Cornell. "The second day, everyone had gone to Jo-Ann fabrics, and had yarn and crochet hooks," said Dr. Taimina. "And these are math professors."
The crossover to the art world began last year. An official of the Institute for Figuring, an educational organization based in Los Angeles, spotted an article about Dr. Taimina's models in New Scientist magazine and invited her and her husband to California to speak about them. An audience that included artists and movie producers was there. "They told us this helps with their imagination," Dr. Henderson said.

In February, the two spoke in New York City. To their surprise, the talk, at the Kitchen, a performance space in Chelsea, sold out. Some enthusiasts asked if they were going on tour.
Gwen Blakley Kinsler, the director of the Crochet Guild of America, believes Dr. Taimina's objects will be of interest to free-form crocheters. "It's always nice to be validated," she said. "People think only grannies do this and it's just doilies."

She plans to publish an article about Dr. Taimina and her hyperbolic creations in Crochet Fantasy magazine later this year.

That would be interesting notoriety for someone who, as a child, was told by her teachers not to waste time in art classes. As an adult, when terrified artists started showing up in her math classes to fulfill their degree requirements, she signed up for a watercolor class, thinking, "Then I will know how they feel."

Now when students tell her they simply cannot understand math, she pulls out one of her paintings and says, "I learned that in three months." Then she might pull out one of her crochet models.