A good counterexample can be worth just as much as a good theorem. A major counterexample about numerous things in mathematics, that is perfect to have in a back pocket if you want to test out a new definition or statement, is the Cantor set.
It is discovered by Henry John Stephen Smith, firstly introduced by German mathematician Georg Cantor, and also named after him.
The Cantor set is deceptively simple example of a set, that possesses number of fascinating properties, which demonstrate that our intuition about space (even such small space as the unit interval) can be wrong. It is an interesting combination of large and small.
In this thesis, it will be defined several variations of a Cantor set and a few di˙erent constructions of it. We will discuss its properties regardless of the construction and variations, but also some that are not necessarily preserved under constructional transformation.
Another thing that we are going to observe, maybe even more fascinating than set itself, is the Cantor function. It is a great counterexample too, for a lot of theorems and definitions in mathematical analysis. It is a continuous function, that looks like everything, but a continuous function; constant but increasing. The point where things get even more interesting is its derivative.
Also known as Devil’s staircase, Cantor function was also introduced by Georg Cantor ans was later popularised by Schee˙er, Lebesgue and Vitali.
Contents
1 Introduction...1
2 Construction of Cantor sets...2
2.1 Cantor ternary set...2
2.2 Base 3 construction...2
2.3 Generalised Cantor λ-set...3
2.4 Generalised Cantor α-set...5
3 Properties of Cantor sets...6
3.1 Topological properties...6
3.2 Measure of Cantor sets...7
4 Cantor ternary function...11
4.1 Construction...11
4.2 Properties...14
4.3 Continuity...17
References...19
1 Introduction
A good counterexample can be worth just as much as a good theorem. A major coun- terexample about numerous things in mathematics, that is perfect to have in a back pocket if you want to test out a new definition or statement, is the Cantor set.
It is discovered by Henry John Stephen Smith, firstly introduced by German mathemati- cian Georg Cantor, and also named after him.
The Cantor set is deceptively simple example of a set, that possesses number of fasci- nating properties, which demonstrate that our intuition about space (even such small space as the unit interval) can be wrong. It is an interesting combination of large and small.
In this thesis, it will be defined several variations of a Cantor set and a few different constructions of it. We will discuss its properties regardless of the construction and variations, but also some that are not necessarily preserved under constructional trans- formation.
Another thing that we are going to observe, maybe even more fascinating than set itself, is the Cantor function. It is a great counterexample too, for a lot of theorems and definitions in mathematical analysis. It is a continuous function, that looks like everything, but a continuous function; constant but increasing. The point where things get even more interesting is its derivative.
Also known as Devil’s staircase, Cantor function was also introduced by Georg Cantor ans was later popularised by Scheeffer, Lebesgue and Vitali.
2 Construction of Cantorsets
One of the best known Cantor sets is Cantor ternary set(or Cantor middle third set). The following points illustrate best its formation:
2.1 Cantor ternaryset
It begins with the interval [0,1], which will be called C0.
Then, we remove the open middle third of C 0 and obtain C 1 = [0, 0 ] ∪ [ 2 ,1].
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Again, from each of the two intervals we are deleting open middle third again.
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By continuing this process, we get C nto be intersection of all C ifor i∈ {0, ..., n}. Every next set is intersection of all the sets so far. Finally, Cantor ternary set is obtained by allowing n to go to infinity.
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2.2 Base 3 construction
Another way of constructing Cantor ternary set is using base 3 expansion of numbers in [0, 1].
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It is clearly seen that points in Cantor ternary set have only 0’s and 2’s in base 3 expansion. Furthermore, we can say that element x∈ [0,1] belongs to the Cantorternary set if and only if its base 3 expansion is x=
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Figure 1: Cantor ternary set
Remark 2.1.If x= 0, a1a2a 3 ...is the base 3 representation of some x ∈ [0,1], then each a ncorresponds to which third xis in. a 1 tells us in which third of [0,1] xbelongs to, a 2 in which third of the given third is it in etc. First third is represented with 0, second with 1 and the third with 2.
Example 2.2.Take x= 0,21 in base 3. By given a 1 = 2 we see that x∈ [ 2 ,1] and
because a= 1 we know that x∈ [ 2 , 8 ], so clearly x∈/C. 3
2 9 9
Remark 2.3.It needs to be cleared that base 3 representation of numbers is not unique.
For example 1 is written in base 3 as 0,1, but also as 0,02222 ...Every real number
can have at most one base 3 expansion using only 0’s and 2’s. All things considered, every point in Cantor ternary set needs to have base 3 expansion using only 0’s and 2’s, although it may also have another.
2.3 Generalised Cantorλ-set
Before we proceed with the definition of the generalised Cantor λ-set, the following ques- tion is imposed: is there anything special about 1 ? To a certain extent, it is in one way, but it is still possible to construct a set with some other number that shares the same topological properties with Cantor ternary set while following the same rules. Lets observe what happens if we construct the set with 1 : We start off again with unit interval C 0 = [0,1].
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Now is open middle fifth removed and we have C 1 = [0, 2 ] ∪ [ 3 ,1].
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Then, from each of those two intervals remove open middle 1 . We now have C 2 =
[0, 5 ] ∪ [ 7 , 2 ] ∪ [ 2 , 23 ] ∪ [ 25 ,1].
5·3
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As we do this infinitely, many times we, obtain Cantor- 1 set:
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Similarly, same construction with 1 is going to look like this:
C 0 = [0,1]
C 1 = [0, 3 ] ∪ [ 4 ,1]
C= [0,74 ] ∪ 75 , 3 ] ∪ [ 4 , 16 ] ∪ [ 17 ,1]
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Like in previous examples, we obtain C7 as infinite intersection of all Cn’s. So, let’s generalise it for any λ∈ (0, 1 ]. Starting, as usual, with the unit interval [0,1], remove open middle interval of length λ. First, open middle interval removed is going to be ( 1 − λ, 1 + λ) to get C1. In the next step remove again open middle interval from each of subinterval of C1, this time of length λto get C2. Continue so, each time removing 2 times as many intervals as in previous step, and each of removed intervals to be one third of a previously removed interval. Any of those sets are topologically the same as Cantor ternary set,so we call them Cantor
λ-sets.
2.4 Generalised Cantorα-set
Some other kind of construction Cantor set would be if one would take even smaller proportions out. If we were to remove some open middle interval, then in the next step we would not remove same interval multiplied by 3, but square it instead we would also get Cantor set topologically the same as Cantor ternary set. Again take some fraction and try it out, 1 for example: First, following the same recipe, we remove open middle fourth from the unit interval
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In the next step, we see the difference between λ-construction and this one. We are not taking 1 out, as we would in λ-construction, we remove 1 instead and get:
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Remove open middle interval of length αfrom [0,1]. This removed interval is the same
as in λconstruction: ( 1 − α, 1 + α). Then remove open middle interval of each of two
segments of length α 2 2 2 2 2 α 3 of each of four segments etc. We continue doing
, then middle
so infinitely many times. What we get is Cantor α-set which is also topologically the same as Cantor ternary set.
In general, we constructed a class of sets with similar properties using similar scaled construction. For this reason, we can call both Cantor λ- and Cantor α-sets Cantor-like sets or just ’a’ Cantor set.
3 Properties of Cantorsets
3.1 Topologicalproperties
• Cantor set(in future use C) in not empty.
Since 0,1 ∈ Cand Ccontains also all endpoints of all removed intervals, Cis not empty. Ccontains actually a lot more, as we are going to see later on.
• Cis closed set.
Cis closed as it is intersection of closed intervals.
• Cis bounded.
Cis bounded because it is a subset of [0,1].
• Cis compact.
We know that using Heine-Borel theorem, which states that a subset of R is com- pact if and only if is closed and bounded.
Definition 3.1.Perfect set is a set that is closed and has no isolated points(every point is accumulation point).
• Cis a perfect set.
For every point in Cwe can always find some E- neighborhood that contains at least some other point in C. There is no E>0 that satisfies C∩(x− E, x+E) = {x}, for x∈ C, so Chas no isolated points. Furthermore, every x∈ C is accumulation
point of C, because it directly follows that C∩ {(x− E,x+ E ) \ {x}} =/ ∅ for x∈ C.
Definition 3.2.Nowhere dense set is a set whose closure has empty interior.
• Cis nowhere dense set.
Since Cis closed, it contains all of its accumulation points. Its closure is
C¯ = C∪ {x∈ R: xis accumulation point of C} =C.
For every x, y∈ Cwe can always find a number between them that requires digit 1 in its base 3 expansion. From that follows that Ccontains no intervals, which mean that its interior is empty.
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Definition 3.3.1) Disconnected set is a set that can be separated into two disjoints sets Aand Bsuch that no accumulation point of Abelongs to Band no accumu- lation point of Bbelongs to A. A set is connected if it is not disconnected.
2) A set is totally disconnected if there is no connected subset that contains more than one point.
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3) Such sets Aand Bare called separated sets and it holds: A∩ B= ∅ and A∩ B= ∅
Theorem 3.4. C is totally disconnected.
Proof.For x, y∈ Cwith x<ywe know that there is no interval (x, y) contained in C, as Ccontains no intervals at all, so there exists z∈ (x, y) \ C. Set A= [0, z) ∩ C and B= (z,1] ∩ C. As we have A= [0, z) ∩ C= [0, z] ∩ Cand B = (z,1] ∩ C= [z,1] ∩ C, we see that Aand Bare closed and open(clopen) in Cwith respect to subspace topology. Closure of both Aand Bare Aand Brespectively and then it is clear that Aand Bare separated because A∩ B = A∩ B= ∅ and A∩ B= A∩ B= ∅, which means that if we can separate Cinto two such sets, Cmust be disconnected.
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Every two points in Care contained in two such disjoint clopen sets, so no two distinct points are in the same connected component of C. Therefore, it follows that all connected component of Cmust be one-point sets(singletons) and that Cis totally disconnected.
Theorem 3.5. Cantor ternary set is uncountable.
Proof.As we can write every x∈ C as x= 0, a1a2a 3 ... in base 3 and a ntells us to which subinterval of Cnxbelongs to, we can switch to binary expression allowing 2’s to become 1’s (as we have no 1’s). We have now 0’s representing left subinterval and 1’s representing right subinterval of some Cn.
Now assume opposite, that Cis countable. Lets list all numbers in C:
x 1 = 0, a11a12a 13 ...x 2 = 0, a21a22a 23 ...x 3 = 0, a31a32a 33 ...
Now define 0 ifa kk = 1
1 ifa kk = 0
y= 0, y1y2y 3 ...is clearly different from all listed numbers, as we defined it so, but it is still in C, because if we switch back from binary to base 3, we would have ywritten using only digits 0 and 2.
That is a contradiction, so it can be concluded that C is in fact uncountable.
Remark 3.6.Every Cantor-like set is uncountable, not just Cantor ternary set. This proof only refers to Cantor ternary set, but one could use similar proof to prove un- countability for any other Cantor set.
To sum it up, it has been seen how big Cantor ternary set is. What follows shows us how small it can also be.
3.2 Measure of Cantorsets
First of all, we should become familiar with measure in general and more precisely Lebesgue measure.
Definition 3.7.Let E⊆ R. We define the (Lebesgue) outer measure on a real line of
E as
k=1
where {Ik} is countable collection of open intervals with E⊂
of interval I.
kL=1
Definition 3.8.1) A set E⊂ R is Lebesgue measurable if for any set A⊂ R
m(A) = m(A∩ E)+ m(A∩ CE)
2) If Eis a Lebesgue measurable set, then the Lebesgue measure of Eis its outer measure m(E) and it is denoted m(E).
Remark 3.9.Lebesgue measure of any interval I⊂ R is its length m(I).
The following two theorems, which will not be proven at the moment, are going to be used later. Proof of both of these theorems can be found in Hewitt’s and Stromberg’s "Real and Abstract Analysis".
Theorem 3.10. Every compact set is Lebesgue measurable.
Theorem 3.11.LetE n be measurable set forn∈ N andE=
1) If E i∩ E j= ∅ for i /= jthen
m(Ei)+ m(Ej) = m(E i ∪ Ej)
2) If E 1 ⊃ E 2 ⊃ ...E n ...then
nn=1
E n bounded.
m(En)
In the following Theorems, we will see what are the measures of Cantor-like sets.
Theorem 3.12. Cantor ternary set is measurable and it has measure zero.
Proof.It follows directly from theorem 3.10. that Cis measurable since it is compact. Using theorem 3.11. we have:
m( Cn) = lim
n→∞
m(Cn)
n=1
Each C
is union of 2nintervals I, each with length t 1 )n. So
n i 3
m(Cn) = m(
i=1
Ii) = 2n· m(Ii) = 2n· 3n= 3
m(C) = m(
nn=1
Cn) = lim
n→∞
m(Cn) = lim
n→∞
2 n= 0
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Is the Lebesgue measure of every Cantor set zero? The answer could be seen in the sets that we have seen so far:
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Proposition 3.13.SetsC5 andC7 are both measurable with positive Lebesgue measure.
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Proof.Both sets C5 and C7 are measurable as they share compactness property with Cantor ternary set.
In order to acquire Lebesgue measure this time we calculate the length of all removed
intervals from unit interval to see what we have left.
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For C5 we firstly removed open interval of length 5 , then we removed 2 open intervals of length 1 . In each step we have removed the interval 3 times smaller than the previous one. So the sum of all length removed is
1 + 2
+ 4
+ ··· =
5 15
1 2 · 1
45
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so it equals: 1
n=0
1
3 = 3 .
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We removed total length of 3 . Subtracting this from length of unit interval, which is 1,
2 5 1
we are left with 5 , so the Lebesgue measure of Cantor 5 -set is
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If we want to try and calculate the Lebesgue measure of any Cantor λ-set, there is nice formula to do so:
Theorem 3.14.The Lebesgue measure of Cantorλ-set ism(C) = (1 − 3 · λ)
1 1
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By subtracting this from 1, we get
2 n
How is the Lebesgue measure of Cantor α-set calculated? Theorem 3.15.The Lebesgue measure of Cantorα-set ism(Cα) = 1−3α
Proof.Adding all intervals removed together we have
α+ 2α 2 + 4α 3 + ··· =
= α(1+ 2α+ 22α 2 + 23α 3 + ··· ) = The sum converges as 2αis always smaller than 1, for αas we defined it:→ 1 for natural
number n>2
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Remark 3.16.We could also obtain the Lebesgue measure of Cantor ternary set this way, subtracting length of all intervals removed and get zero.
Moreover, by equating both of these formulas to zero, we can see that the equation holds only for λ= α= 1 and therefore Cantor ternary set is the only Cantor set which is constructed this way, resulting in zero Lebesgue measure.
Remark 3.17.There are some other constructions which provide Cantor sets other than Cantor ternary set, having zero Lebesgue measure.
4 Cantor ternaryfunction
4.1 Construction
In this chapter we move to the important issue resulting from Cantor set, just as equally interesting, Cantor function.
Like the Cantor set, this function is constructed step by step. First, we begin with identity function that we are going to call g0.
g 0 : [0,1] −→ [0,1]
g(x) = x
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Consider the function g 1 : [0,1] −→ [0,1] which takes constant value 1
on the interval
( 1 , 2 ), the interval we remove from C
2
C. On
3 3 0 as we construct Cantor ternary set to get 1
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So for the first step in constructing Cantor ternary set, where we got set C1, we can make a function that maps every point not in the Cantor set at this particular moment to the constant value 1 .
The next function we are observing is g 2 : [0,1] −→ [0,1]. The function g 2 = g 1 for
x∈ ( 1 , 2 ), but it takes additional constant values 1 and 3 on the intervals ( 1 , 2 ) and
( 7 , 8 3 3
4 4 9 9
CCin
9 9 ) respectively. Again, these are intervals that we remove from 1 to get 2
process of constructing Cantor ternary set. We can again define g 2 to be linear for every
x∈ C2.
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The same procedure is going to be done in the same manner. Each function g nis con- stant on corresponding open intervals at n-th stage of construction of Cantor ternary set - on intervals we removed from set Cn− 1 to get set Cn, and etc.
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As the process progresses, we have a sequence of functions g0, g1, g2,··· , that converges exactly to Cantor function. So let’s define it in the following way:
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What this function is basically doing is changing all the points from [0,1] from their ternary expansion into binary and it is doing so in two ways for two distinct sets.
For point in Cit changes all 2’s into 1’s and read it as binary.
For point in [0,1] \ Cit firstly deletes everything after 1 into 0’s, then again changes all 2’s into 1’s and read it as binary.
What is going on exactly? In ternary expansion of numbers in [0,1] we classified numbers depending on their digits. As we have 3 digits: 0, 1 and 2, each representing in which subinterval of [0,1] number belongs to, as we did it earlier, we can see if our number lies on the left, middle or right subinterval of [0,1]. Left and right subintervals are exactly the points in Cantor ternary set, they do not have any 1’s in their base 3 expansion, so we simply change 2’s to 1’s and read it as binary. It can be written as:
f(0, c1c2c 3 ·· ·(3)) = 0, c 1 , c 2 , c 3 ,···
for
2 2 2
(2)
x= 0, c1c2c 3 ··· ∈ C
For x/= C, xmust have 1 somewhere in its base 3 expansion. These points lie always on middle intervals, so we want to map those to one constant value and that is the reason
why we change every digit after 1 to 0, because we do not care in which subinterval of removed interval point belongs to. So long nunber belongs to removed interval, we stop there and map it to a constant value.
For example, any cwith 1 < c < 2 can be written as c= 0,1c1c2c 3 ··· . We change all
the c ,c ,c ,···
3 3 ,1 is mapped to 1 .
1 2 3
to zero and the result 0 2
When it comes to uniquness of base 3 expansion, function fis well defined. Although there can be more than one ternary representation of some x∈ [0,1] , all non-unique representation of xare mapped to the same number.
For example: 1
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We could also expand fon a whole R letting all numbers in domain smaller than zero to map to zero, and all numbers in domain bigger than one to map to one. We can denote such function as F.
4.2 Properties
0 ifx∈ (−∞,0)
F= fif x∈ [0,1]
⎪⎩ 1 ifx∈ (1 + ∞)
• Cantor function is locally constant.
fis locally constant on the complement of the Cantor ternary set: [0,1] \ C.
• Cantor function is differentiable almost everywhere.
fis differentiable on [0,1] \ C, since it is constant there. Because [0,1] \ Cis a subset of [0,1] with measure 1, which is basically almost everything of the unit interval, we say that fis differentiable almost everywhere.
• Derivative of fvanishes (f1= 0) everywhere where fis differentiable.
It is clear that if fhas derivative, it has to be 1, since derivative exists only on [0,1] \ C, exactly on constant parts of f.
• Cantor function is increasing(non-decreasing).
Although being constant almost everywhere and therefore having no vertical progress almost everywhere, fmanages to rise from 0 to 1, which can be easily seen from its graph. So, fis increasing(non-decreasing), but not strictly increasing: for all x, y ∈ [0,1] with x≤ yone has f(x) ≤ f(y).
Figure 2: Cantor function
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• Arc length of Cantor function is two.
We define now γC(t) = (t, f(t)) as Cantor curve and determine its length. First, lets observe the length of the parts of the Cantor curve where tbelongs to the complement of the Cantor ternary set, so everywhere where fis constant. Sum of lengths of these portions is the same as the length of [0,1] \ C, which we know is 1.
3n
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And therefore, the length of Cantor curve on a [0,1] \ C, isl(γ[0,1]\C) = 1
To calculate the remaining length of the curve, we need the length of the sloped portions of f(x). At n-th stage the function g nhas 2nsuch parts(the same as
number of closed intervals contained in C), and they increase function for 1 overnn
3n a length of 1 .
Now we have the length of the sloped part of some gncurve as:
l(γn”sloped”) = lim (2
n 1 2
+ 1 2
) = lim (
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The arc length of some g nis l(γn) = 1 + 1 = 2 and the length of our Cantor curve would be l(γC) ≥ l(γn) = 2. It has to be bigger that the length of g nbecause the graph of fis even more cut up in vertical and horizontal peaces than gn.
On the other hand , if we go from (0,0) to (1,1), first horizontally, then vertically, we would have total length of 2, which is surely longer that the graph of γC.
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4.3 Continuity
Looking at the graph, it is hard to tell if the Cantor function is continuous or not. For most "well behaved" functions we can tell if the function is continuous if "a graph can be drawn without lifting the pencil from the paper". This naive statement fails to be true for the Cantor function. Although the graph looks broken up into pieces and we could never draw it by hand, the Cantor function is indeed continuous, but not just that, it is uniformly continuous.
Theorem 4.1. Cantor function is continuous
Proof.Every constant function is continuous, so fis continuous on [0,1] \ C.
For all x∈ Cwe want to show that for any E> 0 there is δ >0 such that if |x− y | <δ
then |f(x) − f(y)| <E.
For given Ewe choose nsuch that 1 < E. Then δ= 1 suffices. Why?
2n
Suppose that , =
3n 1 . Then their ternary
x, y∈ Cx/ yare separated by no more than 3n
expansions are the same uo to the first n− 1 digits, and they differ at n-th digit.
x= 0, a1a 2 ··· an−1an
y= 0, b1b 2 ··· bn−1bn
So But
a 1 = b1, a 2 = b2,··· an− 1 = bn− 1 a n/= bn
As we apply fon xand y, as we defined it earlier, we have
f(x) = 0, a1a 2 ··· an−1an
2 2 2 2
f(y) = 0, b1b 2 ··· bn−1bn
2 2 2 2
We see that f(x) and f(y) in their binary expansion are same up to the n− 1 digit,
2
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By looking at the domain of Cantor function, which is interval [0,1] and knowing that continuous functions on a compact sets are uniformly continuous we see that the Cantor function is also uniformly continuous, what was previously stated.
Now, lets go step further in our observation of continuity of Cantor function and see if it is absolutely continuous.
Theorem 4.2. Cantor function is not absolutely continuous
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If we were to assume that fis absolutely continuous, then we would have two conditions
that need to be fulfilled:
• fis differentiable almost everywhere with Lebesgue integrable derivative on [0,1], and
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References
[1] Sterling K. Berberian, Fundamentals of real analysis. Springer, 1999
[2] Diomedes Barcenas, The Fundamental Theorem of Calculus for Lebesgue Integral .
Divulgaciones Matematicas, 2000
[3] Adrian Constantin, Fourier analysis : Part 1 : Theory. Cambridge University Press, 2016
[4] Roberto DiMartino; Wilfredo O. Urbina, Excursions on Cantor-like Sets . Cornell University Library, 2014
[5] O. Dovgosheya; O. Martiob; V. Ryazanova; M. Vuorinenc, The Cantor function .
Expositiones Mathematicae, 2005
[6] Bernard R. Gelbaum; John M. H. Olmsted, Counterexamples in analysis . Holden-
Day, 1966
[7] Edwin Hewit ; Karl Stromberg , Real and abstract analysis : a modern treatment of the theory of functions of a real variable . Springer, 1965
[8] Md. Jahurul Islam; Md. Shahidul Islam, Lebesgue Measure of Generalized Cantor Set . Department of Mathematics, University of Dhaka, Dhaka, Bangladesh, 2015
Frequently asked questions
What is the Cantor ternary set and how is it constructed?
The Cantor ternary set, denoted as C, is constructed starting with the interval [0,1]. The open middle third (1/3, 2/3) is removed, resulting in two intervals: [0, 1/3] and [2/3, 1]. This process is repeated on each remaining interval, removing the open middle third each time. The Cantor set is the intersection of all these sets as the process continues infinitely. It can also be constructed using the base 3 expansion of numbers in [0, 1], where elements of the Cantor set have only 0s and 2s in their base 3 expansion.
What are generalized Cantor λ-sets and α-sets?
Generalized Cantor sets are variations of the Cantor set constructed by removing a middle interval of length λ (Cantor λ-set) or α (Cantor α-set) at each stage. For the Cantor λ-set, at each step an open middle interval of length λ is removed, with λ ∈ (0, 1/3]. Subsequent intervals removed are scaled by a factor of 3. For the Cantor α-set, at each step an open middle interval of length α is removed. Subsequent intervals removed are of length α2, then α3, and so on.
What are the topological properties of Cantor sets?
Cantor sets, including the Cantor ternary set and its generalizations, have several key topological properties:
- Non-empty
- Closed
- Bounded
- Compact
- Perfect (every point is an accumulation point)
- Nowhere dense (its closure has an empty interior)
- Totally disconnected (no connected subset contains more than one point)
- Uncountable
What is the Lebesgue measure of Cantor sets?
The Lebesgue measure of the standard Cantor ternary set is zero. However, generalized Cantor sets (λ-sets and α-sets) can have positive Lebesgue measure depending on the length of the intervals removed. The Lebesgue measure of a Cantor λ-set is given by m(C) = (1 − 3λ), and of Cantor α-sets given by m(Cα) = (1 − 2α)/(1 - α). Only for Cantor ternary set, constructed by removing intervals of length 1/3, resulting in zero Lebesgue measure.
What is the Cantor ternary function (Devil's staircase) and how is it constructed?
The Cantor ternary function, also known as the Devil's staircase, is a function constructed step-by-step. It starts with the identity function g0(x) = x on [0, 1]. At each stage, it takes a constant value on the middle third interval removed during the construction of the Cantor set. The limit of these functions as the process continues is the Cantor function. It can be described by changing the points from [0,1] from their ternary expansion into binary.
What are the properties of the Cantor ternary function?
The Cantor ternary function has the following properties:
- Locally constant on the complement of the Cantor set
- Differentiable almost everywhere, with a derivative of 0 where it is differentiable
- Increasing (non-decreasing)
- Continuous (and uniformly continuous)
- Arc length of two
- Not absolutely continuous
Is the Cantor function continuous?
Yes, the Cantor function is continuous. It is continuous on [0,1] \ C, the complement of the Cantor ternary set. Also, since it’s a continuous function on compact sets, Cantor function is uniformly continuous.
Is the Cantor function absolutely continuous?
No, the Cantor function is not absolutely continuous, since its derivative vanishes almost everywhere and f(1) − f(0) /= int from 0 to 1 of f'(t)dt.
- Quote paper
- Jovana Roljic (Author), 2018, On Cantor sets, Munich, GRIN Verlag, https://www.grin.com/document/512831
