Intereting Posts

Quintic diophantine equation
A Math function that draws water droplet shape?
Infinite Series $\sum\limits_{n=1}^\infty\frac{H_{2n+1}}{n^2}$
About the definition of Cech Cohomology
What is an intuition behind total differential in two variables function?
Question on conservative fields
there exist some real $a >0$ such that $\tan{a} = a$
${a_n}$ series of Fibonacci numbers. $f(x)=\sum_{0}^{\infty}a_nx^n$, show that in the convergence radius: $f(x)= \frac{1}{1-x-x^2}$
Find the Mean for Non-Negative Integer-Valued Random Variable
Geometric interpretation of connection forms, torsion forms, curvature forms, etc
Show that the linear transformation $T:V\to V$ defined by $T(B)=ABA^{-1}$ is diagonalizable
Usage of dx in Integrals
A smooth function's domain of being non-analytic
Find the last two digits of the number $9^{9^9}$
question about Laguerre polynomials

This was a problem from a class that I thought was really interesting. It asked to find function $f\in C[0,1]$ such that the sets $\{x:f(x)=c\}$ form a Cantor Set for all $0\leq c\leq 1$. I found a non-constructive proof of the existence of such functions, but would be curious if anyone could give a constructive example of such a function.

**Definition:** A set is a Cantor Set if it is uncountable, perfect, compact, nowhere-dense set. This makes it homeomorphic to “the Cantor set.”

- What is a simple example of a limit in the real world?
- arbitrary large finite sums of an uncountable set.
- Importance of Least Upper Bound Property of $\mathbb{R}$
- $f$ is continuous at $a$ iff for each subset $A$ of $X$ with $a\in \bar A$, $f(a)\in \overline{ f(A)}$.
- A derivation of the Euler-Maclaurin formula?
- Closed form for definite integral involving Erf and Gaussian?

- For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous.
- Uniform continuity on $$ and $ $ $\implies$ uniform continuity on $$.
- Second derivative positive $\implies$ convex
- Integral in $n-$dimensional euclidean space
- Closed form for definite integral involving Erf and Gaussian?
- Check my answer: Prove that every open set in $\Bbb R^n$ is a countable union of open intervals.
- $\sqrt{x}$ isn't Lipschitz function
- Proving that $f(n)=n$ if $f(n+1)>f(f(n))$
- Proof Verification : Prove -(-a)=a using only ordered field axioms
- Proof of continuity of Thomae Function at irrationals.

As the Cantor set is self-similar, it makes sense to try to construct a graph with some kind of self-similar structure. The only way the graph of a function can be *strictly* self-similar is if it is a line segment. Thus, we explore a slight generalization using a self-affine graph. I’ll first present an example that is specifically designed to have the desired property and then comment on a couple of cases where such a structure arises quite naturally.

For our first example, we start with the Iterated Function System of six affine functions defined as follows:

$$

\begin{align}

f_1(x,y) &= \left(\begin{array}{cc} 1/6 & 0 \\ 0 & 1/4 \end{array}\right)

\left(\begin{array}{c} x\\y\end{array}\right) \\

f_2(x,y) &= \left(\begin{array}{cc} 1/6 & 0 \\ 0 & 1/4 \end{array}\right)

\left(\begin{array}{c} x\\y\end{array}\right) +

\left(\begin{array}{c} 1/6\\1/2\end{array}\right) \\

f_3(x,y) &= \left(\begin{array}{cc} 1/6 & 0 \\ 0 & -1/4 \end{array}\right)

\left(\begin{array}{c} x\\y\end{array}\right) +

\left(\begin{array}{c} 2/6\\1\end{array}\right) \\

f_4(x,y) &= \left(\begin{array}{cc} 1/6 & 0 \\ 0 & -1/4 \end{array}\right)

\left(\begin{array}{c} x\\y\end{array}\right) +

\left(\begin{array}{c} 3/6\\1/2\end{array}\right)\\

f_5(x,y) &= \left(\begin{array}{cc} 1/6 & 0 \\ 0 & 1/4 \end{array}\right)

\left(\begin{array}{c} x\\y\end{array}\right) +

\left(\begin{array}{c} 4/6\\0\end{array}\right)\\

f_6(x,y) &= \left(\begin{array}{cc} 1/6 & 0 \\ 0 & 1/4 \end{array}\right)

\left(\begin{array}{c} x\\y\end{array}\right) +

\left(\begin{array}{c} 5/6\\1/2\end{array}\right)

\end{align}

$$

The geometric action of this IFS on the upwardly oriented unit square is illustrated in the following figure:

This image shows the oriented initial set, the first two levels of approximation using rectangles, and a higher order approximation to the attractor, which is the graph of a continuous function.

Now, it’s pretty easy to see that the intersection with the level $n$ approximation by rectangles with a horizontal line segment is just a collection of line segments and that these contain the corresponding intersection at level $n-1$. The intersection of all these is perfect, non-empty, and no where dense. Thus, it’s a Cantor set.

This function maps $[0,1]\to[0,1]$ but it’s easy to extend to $\mathbb R$, if desired.

There are some examples of functions with Cantor type level sets that occur naturally – i.e. constructed for some other purpose but happen to have the desired property.

The coordinate functions of Hilbert’s space filling curve have Cantor type level sets. I proved this in my paper The Hausdorff Dimension of Hilbertʼs Coordinate Functions. I also published a more elementary exposition in The Math Mag. If you understood the previous example, you can see what’s going on in figure 7 of that paper:

Finally, the Takagi function has this property at many points. More specifically, let

$\varphi(x)$ denote the distance from $x$ to the nearest integer and let

$$\tau(x) = \sum_{k=0}^{\infty} \frac{1}{2^k}\varphi(2^kx).$$

Then, the maximum value of $\tau$ is $2/3$ and $\tau^{-1}(2/3)$ is a self-similar Cantor set with dimension $1/2$. The partial self-similar structure of the Takagi function implies that the level sets are Cantor sets for a dense set of points in $[0,2/3]$.

This an many other fun facts are proved in The Takagi function: a survey. It’s not hard at all to see what’s going on by examining the even partial sums of the series that generates $\tau$:

The image shows

$$\tau(x) = \sum_{k=0}^{n} \frac{1}{2^k}\varphi(2^kx)$$

for the first few even values of $n$. Note that the top is always a set of segments. As we move from $n=2$ to $n=4$, the one segment is replaced by two smaller sub-segments. The partial self-similar structure of the object suggests that this pattern repeats and, in the limit, the top of the object is a Cantor set.

Again referring to the partial self-similar structure, there is a dense set of points in $[0,2/3]$ whose inverse image is a Cantor set. This is formalized in Lemma 3.3 of the linked paper. Specifically,

the portion of the graph of $\tau$ above the interval

$[k/2^{2m},(k+1)/2^2m]$ is a

similar copy of the full graph reduced by the factor

$1/2^{2m}$ and shifted up by $\tau(k/2^{2m})$.

- Adding Elements to Diagonal of Symmetric Matrix to Ensure Positive Definiteness.
- How to use the Extended Euclidean Algorithm manually?
- How to calculate the derivative of this integral?
- How to find continued fraction of pi
- Prove: The positive integers cannot be partitioned into arithmetic sequences (using Complex Analysis)
- How to prove that $f(A)$ is invertible iff $f$ is relatively prime with the minimal polynomial of $A$?
- Will $2$ linear equations with $2$ unknowns always have a solution?
- Fourier transform of Schrödinger kernel: how to compute it?
- Is there an accepted term for those objects of a category $X$ such that for all $Y$, there is at most one arrow $X \rightarrow Y$?
- Volterra Operator is compact but has no eigenvalue
- Prove that $x-1$ is a factor of $x^n-1$
- Does an equivalence of $G$-sets and $H$-sets imply an isomorphism of $G$ and $H$?
- Interspersing of integers by reals
- Inverse image sheaf and éspace étalé
- Induction for statements with more than one variable.