Question

There are many "strange" functions to choose from and the deeper you get involved with math the more you encounter. I consciously don't mention any for reasons of bias. I am just curious what you consider strange and especially like.

Please also give a reason why you find this function strange and why you like it. Perhaps you could also give some kind of reference where to find further information.

As usually: Please only mention one function per post - and let the votes decide :-)

Answer 1

A Brownian motion sample path.

These are about the most bizarrely behaved continuous functions on $\mathbb{R}^+$ that you can think of. They are nowhere differentiable, have unbounded variation, attain local maxima and minima in every interval... Many, many papers and books have been written about their strange properties.

Edit: As commented, I should clarify the term "sample path". Brownian motion is a stochastic process $B_t$. We say a sample path of Brownian motion has some property if the function $t \mapsto B_t$ has that property almost surely. So, run a Brownian motion, and with probability 1 you will get a function with all these weird properties.

Answer 2

The formula for the nth term in the Fibonacci sequence

$F_{n} = \cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1+\sqrt{5}}{2}\right)^n-\cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1-\sqrt{5}}{2}\right)^n$

This is interesting because it is a non-recursive expression for the Fibonacci sequence and also because it involves the golden ratio.

Answer 3

I like the beauty and mysticism of Euler's identity:

$$f(\theta) = e^{i\theta} = \cos\theta + i \sin\theta$$

Answer 4

The Conway base 13 function has to be the weirdest function I know. This function is continuous nowhere, yet it satisfies the intermediate value theorem. Only John Conway...

Answer 5

Nonconstant continuous locally recurrent functions are quite unintuitive. A real-valued function is locally recurrent on $\mathbb R$ if for every $x_0\in\mathbb R$ and every deleted neighborhood $N(x_0)$ of $x_0$, there exists $x\in N(x_0)$ for which $f(x)=f(x_0)$. Thus in some sense a nonconstant continuous locally recurrent function looks everywhere like $x\sin(1/x)$ at $x=0$. See papers in the American Math. Monthly of Bush (1962), Marcus (1963), and Mauldon (1965).

Answer 6

Thomae's function, also called the "popcorn function". It's continuous at all irrationals and discontinuous at all rationals. Here a picture:

Answer 7

The function defined by the power series $f(x)=x-x^2+x^4-x^8+x^{16}-\dots$ What is its limit as $x$ approaches $1$ from below? EDIT (This answer is a trick question.)

Answer 8

How about the function given by the Banach-Tarski paradox? This maps a ball into two copies of the same size ball, and is composed of isometries on subsets of $\mathbb{R}^3$.

Answer 9

How about a function f: f(f(x)) = exp(x).

Answer 10

I can't believe no one has mentioned the Dirichlet function:
(I guess it's up to me to bring it up...)

Answer 11

The Fabius function

Answer 12

Just a simple construction to illustrate Nate Eldredge's answer about functions with dense graphs. Pick any $\mathbb{R}$-vector space E with a norm. On E, choose a non-continuous linear form $L: E \to \mathbb{R}$; now this can only be done if $\dim(E)=\infty$, of course.

Then, pick y such that $L(y)=1$, and let $T: E \to E$ be defined by $Tx=x-L(x)y$. Then obviously T maps E onto the kernel of L; it is not difficult to prove that $\ker (L)$ must be dense in E for any non-continuous L (the two conditions are even equivalent), and thus the graph of T must be dense in $E \times E$.

Answer 13

Characteristic p commutative algebra leads naturally to the construction of various continuous functions on [0,1]^m that have beautiful self-similarity properties; for explication and some pictures see:

Pedro Teixeira, Syzygy gap fractals--I, arXiv 1008.0583

Answer 14

Any of the isomorphisms $\mathbb{C}'\to S^{1}$, where $S^{1}$ is the unit circle and $\mathbb{C}'$ is the non-zero complex numbers, with the group operation for both being multiplication.

Answer 15

Fix a probability $p < 1/2$ of winning an unfair coin toss. For $x \in [0,1]$ rational, let $f(x)$ be the probability that, if you started with $x$ dollars, you could make it to 1 dollar through optimal betting* on the outcome of these coin flips. This function $f(x)$ is obviously weakly increasing on $[0,1]$ (in fact strictly). Less obvious is that it extends to a continuous function on $[0,1]$, whose derivative exists almost everywhere, but that derivative is $0$.

http://www.maa.org/joma/Volume8/Siegrist/RedBlack.pdf

Answer 16

The Banach limit assigns to every bounded sequence of real numbers a real number "limit" in a way that is linear, shift invariant, and agrees with the usual limit whenever it exists. Banach limits are among the mysterious examples of continuous linear functionals on $\ell^\infty$ that aren't represented by elements of $\ell^1$. Unfortunately, the Hahn-Banach theorem is used in the construction of the Banach limit, and the values aren't canonical. There's a precise definition at http://en.wikipedia.org/wiki/Banach_limit .

Answer 17

These functions like the Cantor function and the continuous-but-not-differentiable function are all well and good, but contrived - the only place you ever see them is as counterexamples. Here is a function that has many uses in Number Theory, and still manages to have a strange property or two. Let $x=h/k$ with $h$ and $k$ integers, $k>0$. Define $$s(x)=\sum_{c=1}^{k-1}((c/k))((ch/k))$$ where $((y))=0$ if $y$ is an integer, $((y))=\lbrace y\rbrace-1/2$ otherwise. It is easily proved that the sum depends only on the ratio of $h$ and $k$, not on their individual values, so $s$ is a well-defined function from the rationals to the rationals. It is known as the Dedekind sum; it came up originally in Dedekind's study of the transformation formula of the Dedekind $\eta$-function.

Now for the strange properties.

Hickerson, Continued fractions and density results for Dedekind sums, J Reine Angew Math 290 (1977) 113-116, MR 55 #12611, proved that the graph of $s$ is dense in the plane.

With Nick Phillips, I proved (Lines full of Dedekind sums, Bull London Math Soc 36 (2004) 547-552, MR 2005m:11075) that, with the exception of the line $y=x/12$, every line through the origin with rational slope passes through infinitely many points on the graph of $s$. We suspect that the points are dense on those lines, though we could only prove it for the line $y=x$.

Answer 18

A canonical example from elementary real analysis - the Blancmange function. Consider $f$ defined piecewise by

$f(x) = x - [x], \quad \text{if} \quad 0 \leq x- [x] \leq \frac{1}{2}$,

and

$f(x) = 1 - (x - [x]), \quad \text{if} \quad \frac{1}{2} < x - [x] < 1$,

(where $[x]$ is the integer part of $x$). Then define the Blancmange function, $B$

$B(x) = \sum_{n=0}^{\infty}\dfrac{1}{2^n} f(2^{n}x)$.

The series converges by the Comparison Test, since $|f(2^{n}x)| \leq \frac{1}{2}$, for all $x \in \mathbb{R}$, and it can be shown that $B$ is uniformly continuous but nowhere differentiable. Here a picture of the function:

A tasty counterexample to the converse of "differentiability $\implies$ continuity".

Answer 19

f(x) = sin (1/x): (x not 0); f(x) = 0 (x equals 0)

Answer 20

Interpreting your questions a bit liberally, I suggest the Goodstein sequence:

http://en.wikipedia.org/wiki/Goodstein%27s_theorem

Answer 21

Since Mariano took my favorite already, I'll go with the stopping time function for the 3x+1 problem: http://www.ieeta.pt/~tos/3x+1.html

Answer 22

The Osgood curve ("A Jordan Curve of Positive Area") is an injective map from [0,1] into $\mathbb{R}^2$ which traces out an image of positive area. (This differs from standard space-filling curves, which are not injective.)

Answer 23

I'm still quite impressed about $f(x)=\mathrm e^x$ …

Answer 24

It is pretty obvious after you've seen it, but I like the crinkled curve from Halmos's Hilbert Space Problem book:

Let $f:\mathbb{R}\rightarrow(0,\infty)$ be an $L^2$ function, and define $t\mapsto g_t:\mathbb{R}\rightarrow L^2(\mathbb{R})$ by $$g_t(x)=\chi_{(-\infty,t)}(x) \times f(x).$$

Then $g_t$ has the property that for all $t_1 < t_2 < t_3$ the secants $g_{t_2}-g_{t_1}$ and $g_{t_3}-g_{t_2}$ are mutually orthogonal. (The curve turns a corner at every point.)

Answer 25

I suppose the strangest function in mainstream mathematics is Riemann zeta function http://en.wikipedia.org/wiki/Riemann_zeta_function

$ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots \;\;\;\;\;\;\;{Re}(s) >1. $

It is part of one of the most important hypothesis and is very influential in many branch of moder mathematics. It is actively used in many areas and is researched in many ways, it is not curiosity, or exotic example, but important mathematical being!

And is mysterious and strange! Take a look:

Answer 26

The Ackermann function $A(n,m)$ is defined on the natural numbers by a very simple recursion, but the values grow enormously, almost beyond conception. This function completely transcends any simple-minded system of rates-of-growth based on polynomial, exponential, double-exponential and so on.

The first few values of the diagonal function $A(n) = A(n,n)$ are:

• $A(0) = 1$
• $A(1) = 3$
• $A(2) = 7$
• $A(3) = 61$
• $A(4) = 2^{2^{2^{65536}}}-3$
• $A(5)$ is vast, and can be described in terms of exponential stacks of $2$s, whose height is a stack of $2$s, etc. 5 times.
• $A(6)$ is so vast, it is best described using the Ackermann function itself.

The levels of the Ackerman function $A_n(m)=A(n,m)$ stratify the primitive recursive functions, in the sense that they are each primitive recursive, but every primitive recursive function is bounded by such a level of the Ackermann function. Thus, the Ackermann function itself is not primitive recursive, although it is computable in the sense of computability theory.

Answer 27

The empty function $\emptyset:\emptyset\to\emptyset$ is quite strange when you first meet it.

Answer 28

One can construct a natural 'metric' for the Riemann sphere which is equivalent to the spherical metric but which is singular on a dense set of points of the Riemann sphere though remains $L^1$ integrable.

These are built from degree 2 rational maps (first constructed by Mary Rees) which have the whole Riemann sphere as their Julia sets, and have the orbits of their critical points also dense. The Carlesson-Jones-Yoccoz construction of a expanding metric for critically-finite rational maps actually extends to this case, and we get a metric in which this Julia set actually looks as if it was hyperbolic!

[The details are worked out in my PhD thesis, never published as I decided that computer algebra suited me better than complex dynamics].

Answer 29

Please excuse me if I include two related functions in one answer. Any space filling curve is rather strange, at least for me. Let $\gamma\colon[0,1]\to[0,1]^2$ be such a curve, that is, $\gamma$ is continuous and surjective. Let $\gamma(t)=(x(t),y(t))$; then $x(t)$ (or $y(t)$) is my other candidate for strangest function: given any $z\in[0,1]$, $x^{-1}(z)$ has the cardinality of the continuum.

Answer 30

I like the Cantor function. A continuous, increasing function $f:[0,1]\rightarrow[0,1]$ with derivative $0$ almost everywhere. See wiki article here.