Favourite pathological examples

[fourier jr]

I mean of course stuff like functions that are continuous everywhere but differentiable nowhere, or a continuous function that has a Fourier series that diverges everywhere. What's everyone's favourite?

 

[snipez90]

Hmmm, well you can generate a bunch of continuous everywhere but nowhere differentiable functions right? I mean I think it was a big deal when Weierstrass first did it, but now there are lots of simpler (counter)examples. A cute exercise for a honors calculus course is to determine and prove the existence of a function which is continuous at exactly one point, and differentiable at that point (maybe the nastiest thing I saw at that point, besides Thomae's function). Also I guess the topologist's sine curve is considered a pathological function in calculus courses, but it turns out that showing the basic topological fact that it is connected but not path connected only requires basic epsilon-delta arguments.

Although this is probably not considered particularly pathological, the p-adic topology on Z is kind of weird. We learned today that with respect to the metric that induces this particular topology, all triangles are isosceles, each open ball is open and closed (and proving closedness is not simply a matter of vacuous truth, as in the case for the discrete metric), and any point in an open ball is the open ball's center. This was interesting, but I find it hard to take this p-adic stuff seriously probably because the other day we were learning about normed linear spaces and now all of a sudden the instructor decided to construct the p-adic number system so yeah.

 

[LCKurtz]

Not pathological, but sure to generate discussion in most any math class, the story of the midget and the pool balls.

A funnel with an infinite sequence of pool balls, all numbered and in the sequence of natural numbers is above a large barrel. At fifteen minutes before noon balls 1 thru 10 instantly drop into the barrel. The midget, who has been dozing and whose job it is to keep the barrel empty is awakened by the noise. He knows his job is to keep the barrel empty and the inspector will arrive at exactly noon.

He is very fast and orderly and good at his job. He sees the balls and removes ball number 1. There is lots of time remaining so no hurry, back to sleep. At 7.5 minutes before noon balls 11 through 20 are added to the barrel and the midget, being very orderly and fast, removes ball number 2 and goes back to sleep.

This continues -- at 3.75 minutes balls 21 through 30 drop in and he removes ball 3.

This continues with the time dividing by 2 each time and throwing out the next ball. It gets very hectic near noon but, hey, this is a thought experiment.

No question that noon happens exactly 15 minutes after this started. Everything stops instantly and the inspector arrives. Does the midget get fired because the barrel has balls in it or commended for doing such a good job of keeping it empty given what has just happened?

 

[fourier jr]

^ hmm... I have a feeling the answer depends on the use of a funnel, instead of a hopper, a pipeline, a robotic arm, conveyor belt, a bucket or some other conveyance

i also like \int_{+\infty}^{+\infty}

I bet a lot of people think wtf?! when they see it for the first time

 

[quasar987]

I've given a talk last week to undergrads about pathological examples. Here are some of them.

There is a function that is strictly increasing and whose derivative vanishes almost everywhere.

Also mind boggling are functions that are nowhere locally bounded and continuous functions that are nowhere monotonous.

It is possible to find a polyhedra inscribed in the cylinder of unit height and unit radius whose area is arbitrarily large.

There exist 3 open connected subsets of the plane that share a common boundary.

There exists a partition of the unit ball into 5 subsets and 5 rigid motions (isometry) of R^3 that take the 5 parts of our ball onto 2 disjoint copies of the unit ball. In laymen terms, there is a way to chop up the unit ball in 5 pieces and then move them around with translations, rotations and reflections and end up with 2 copies of your original ball. This is a special case of Banach-Tarski's paradox. Another case states that you can chop up a pea in 5 pieces and reassemble them into the sun.

There is a smooth function that has a minimum at 0 but whose derivative there does not pass from negative to positive.

Pretty amazing is the existence of a continuous surjection from [0,1] to [0,1]², and more generally from [0,1] to [0,1]^n (space filling curves). There are space filling curves that are nowhere differentiable. Others are almost everywhere differentiable. This also implies that |R|=|R^n|.

Koch's snowflake is a simple closed curvewith the property that it encloses a finite area while having an infinite lenght. In fact , between any two of its points, it has infinite lenght.

I find interesting the fact that the surface of revolution obtained by rotation the function f:[1,infty]-->R, f(x)=1/x around the x-axis has infinite area but finite volume. Meaning that you can fill the resulting trumpet with paint but you cannot paint the thing!

A continuous strictly positive R-->R map that is unbounded at infinity but whose integral over R is finite.

 

[fourier jr]

in every course there seem to be at least a couple things that are very counterintuitive, or surprising or weird or wacky, etc. in topology there are sure to be many more. i think i remember that in a non-Hausdorff space any sequence can converge to anything, like the sequence 1, 1, 1,... can converge to pi, etc. in differential equations I remember expanding exp(A) in a power series with a matrix A...