# Cantor Set and Function

A continuous function may grow considerably virtually without changing.

The function with this property is easily constructed on the *Cantor set C _{0}*. So I'll
proceed in two steps. First we'll look into C

_{0}and then define the promised function.

When I was a freshman, a graduate student showed me the Cantor set, and remarked that although there were supposed to be points in the set other than the endpoints, he had never been able to find any. I regret to say that it was several years before I found any for myself. |

Ralph P. Boas, Jr |

# Cantor set C_{0}

First of all C_{0} is a subset of the closed unit interval _{0} is what's left over after the removal of a sequence of open subintervals of [0, 1]. The algorithm is as follows:

- Divide the remaining intervals each into three equal parts.
- Remove the open middle interval.
- Repeat 1.

Thus first we remove the open interval (1/3, 2/3). This leaves a union of two closed intervals: _{1,1}. Next we split each of the remaining intervals _{2, 1}= (1/9, 2/9)_{2,2} = (7/9, 8/9)

Observe that we started with the interval [0, 1] of length 1. After removing d_{1,1}, the total length of the remaining intervals became 2/3. d_{2,1} and d_{2,2} each contributed 1/3 to the total. So after their removal, the four remaining intervals had the total length of (2/3)^{2}. Next we remove 4 middle intervals _{3,1}, ..., d_{3,4}^{3}. The process never stops. In general, on the step number *p* we remove 2^{p-1} intervals_{p,1}, ..., d_{p,2p-1}.^{p}.
Obviously, as *p* grows, the length (2/3)^{p} tends to 0. However, this does not mean that C_{0} is empty. Moreover, the set is not even countable. The most convenient way to see this is by using ternary representation of the decimals from

For example _{10} = (0.1)_{3},_{10} = (0.2)_{3},_{10}3,_{10} = (0.02)_{3},_{10} = (0.21)_{3}, and _{10} = (0.22)_{3}.

C_{0} = {x: x = (0.a_{1}a_{2}a_{3}...)_{3}, where each a_{j} = 0 or 2}.

Applying the diagonal process we immediately see that C_{0} is not countable.

Thus, this is the Cantor set C_{0} which has many remarkable properties. For example, the Cantor set has no *isolated* points. In other words,
every neighborhood of every point in C_{0} contains infinitely many other points from C_{0}. (A point of a set is *isolated* if in some of its neighborhoods there are no other points from that set.) This makes C_{0} a *perfect* set (since it is obviously closed.) I'll use C_{0} as a basis for the construction of the function announced at the beginning of the page.

## Remark

1/4∈C

_{0}. Indeed, (0.25)_{10}= (0.020202...)_{3}. Similarly, since(0.75) 3/4∈C_{10}= (0.202020...)_{3},_{0}. The reciprocal of 13 also happens to be sufficiently "lucky" to be a member of C_{0}:1/13 = (0.002002002...) _{3}.A similar construction yields Cantor sets of positive measure.

## Cantor function

Let x = (0.a_{1}a_{2}a_{3}...)_{3}∈C_{0} which means that all a_{i}'s are either 0 or 2. Let define a function _{0}→[0, 1]

let b_{i} = a_{i}/2, then define f(x) = (0.b_{1}b_{2}b_{3}...)_{2}, where x = (0.a_{1}a_{2}a_{3}...)_{3}.

What's interesting is that at the end points of the intervals d_{p,k} so defined function takes on equal values. Consider, for example, _{1,1} = (1/3, 2/3).

f(1/3) = f(0.0222...) = (0.0111...)_{2} = (0.1)_{2} = 1/2.

On the other hand, f(2/3) = f(0.2) = (0.1)_{2}. Therefore _{0}. Using the feature we just mentioned, we can extend (lift) the definition of f to the whole _{p,k}. Since C_{0} has no
isolated points, and is *monotonous* on C_{0}, the extended function is actually continuous being constant on intervals of total length 1!. In other words, a nonconstant function f that has a derivative equal to 0 on intervals of total length 1 manages to grow from 0 to 1 on the interval [0, 1]. The derivative is, of course, discontinuous; it's undefined at the points of the Cantor set C_{0}.

## Remark 1

Cantor function is as famous as it is useful for other exceptional constructs. For example,
let *almost everywhere.*

## Remark 2

Cantor's function serves a very specific example for *Lebesgue's Theorem*

Every monotone function has a well defined finite derivative almost everywhere.

Cantor's function is monotone increasing and is also continuous - a condition not required by Lebesgue's Theorem. It has a derivative everywhere except at Cantor's set which is a set of (Lebesgue) measure zero. This explains the term "almost everywhere".

## References

- B.R.Gelbaum and J.M.H.Olmsted,
*Counterexamples in Analysis*, Holden-Day, 1964 - B.R.Gelbaum and J.M.H.Olmsted,
*Theorems and Counterexamples in Mathematics*, Springer-Verlag, 1990

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