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Is there an easy way to see that the Cantor function is not absolutely continuous that directly uses the definition of absolutely continuous?

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5 Answers 5

up vote 14 down vote accepted

A function $f: E \to \mathbb{R}$ is absolutely continuous on an interval $E$ if for every $\epsilon > 0$ there is a $\delta > 0$ such that whenever a finite sequence of pairwise disjoint sub-intervals $(x_k, y_k)$ of $E$ satisfies

$$ \sum_{k} |y_{k} - x_{k}| < \delta$$

then

$$\sum_{k} |f(y_{k}) - f(x_{k})| < \epsilon$$

Put in words, an absolutely continuous function does not fluctuate on a set of measure zero. To see that the Cantor function is not absolutely continuous, pick $\epsilon < 1$. Then, for every $\delta > 0$, I can find a collection of intervals $(x_{k},y_{k})$ that cover the Cantor points in $[0,1]$ such that

$$ \sum_{k} |y_{k} - x_{k}| < \delta$$

this is because the Cantor set has measure zero. However, since the Cantor function only changes on the Cantor set,

$$\sum_{k} |f(y_{k}) - f(x_{k})| = 1$$

and absolute continuity is violated.


More generally, note that the Cantor function is singular. It is easy to prove (and feels right intuitively) that an absolutely continuous, singular function must be constant. However, the Cantor function is far from constant.

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2  
Excellent answer –  Daniel Montealegre Mar 28 '12 at 6:55
    
The answer taken is not correct since one can not cover canter set by finite union of sets with arbitrarily small measure in total. One can check Royden (4ed) P120 for reference to this question. And the solution taken will become true (with some modification) after proveing Prob38 on P123. –  user49021 Nov 11 '12 at 16:33
1  
The Cantor set $C$ is the intersection of sets $C_n$ constructed iteratively. We know that $C_n \supset C_{n+1}$. Moreover, the total length of the line segments in $C_n$ is $(2/3)^n$. Note that a cover for any $C_n$ is automatically a cover for $C$. Do you not think that I can take $N$ large enough to make $(2/3)^N$ smaller than some fixed $\delta$? –  Isaac Solomon Nov 11 '12 at 18:44

The variation of the Cantor function on each approximation $C_n$ of the Cantor set is $1$. The measure of $C_n$ goes to zero when $n\to\infty$. Ergo.

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One definition of absolutely continuous functions is that they map sets of measure zero to sets of measure zero. However, the cantor ternary function maps the cantor set (of measure zero) onto $[0,1]$.

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Let $f$ the Cantor function. $f$ is increasing not negative, $f(0)=0$ and $f(1)=1$. Then $f$ is differentiable a.e. and since $f$ is constant on every interval removed in the construction of the Cantor set, $f^\prime=0$ a.e.

Once we assume that $f$ is absolutely continuous, we have $$\int_0^1 f^\prime=f(1)-f(0),$$ but this says that $0=1$. Thus $f$ can not be absolutely continuous.

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Potential typo: "uniformly" should probably be "absolutely" (both times). –  Quinn Culver Jun 11 '12 at 5:57

Since Lipschitz's condition is absolutely continuous and cantor function does not satisfies lipschitz's condition.It follows that cantor function is not absolutely continuous.

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How does that follow? There are absolutely continuous functions that are not Lipschitz, for example $\sqrt{x}$ on $[0,1]$. –  Martin Jan 30 '13 at 13:23

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