Difference between revisions of "Cantor ternary function"
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− | The Cantor ternary function (also called Devil's staircase) is a continuous monotone function $f$ mapping the interval $[0,1]$ onto itself, with the remarkable property that its derivative vanishes almost everywhere (recall that any monotone function is differentiable almost everywhere, see for instance [[Function of bounded variation]]). | + | The Cantor ternary function (also called Devil's staircase and, rarely, Lebesgue's singular function) is a continuous monotone function $f$ mapping the interval $[0,1]$ onto itself, with the remarkable property that its derivative vanishes almost everywhere (recall that any monotone function is differentiable almost everywhere, see for instance [[Function of bounded variation]]). |
− | The function can be defined in the following way ( | + | {{:Cantor ternary function/Fig1}} |
+ | |||
+ | The function can be defined in the following way (see, for example, Exercise 46 in Chapter 2 of {{Cite|Ro}}). Given $x\in [0,1]$ consider its ternary expansion $\{a_i\}$, i.e. a choice of coefficients $a_i\in \{0,1,2\}$ such that | ||
\[ | \[ | ||
− | x = \sum_{i= | + | x = \sum_{i=1}^\infty \frac{a_i}{3^i}\, . |
\] | \] | ||
Define $n(x)$ to be | Define $n(x)$ to be | ||
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Then | Then | ||
\[ | \[ | ||
− | f(x) = \sum_{i= | + | f(x) = \sum_{i=1}^{n(x)-1} \frac{a_i}{2^{i+1}} + \frac{1}{2^{n(x)}}. |
\] | \] | ||
For alternative definitions see Example 1.67 of {{Cite|AFP}} and page 55 of {{Cite|Co}}. | For alternative definitions see Example 1.67 of {{Cite|AFP}} and page 55 of {{Cite|Co}}. | ||
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It follows trivially from the definition that $f$ is locally constant on the complement of the [[Cantor set]] $C$: since the Cantor set is a set of Lebesgue measure zero, the derivative of $f$ vanishes almost everywhere. For the same reason, the [[Generalized derivative|distributional derivative]] of $f$ is a [[Singular measures|singular measure]] $\mu$ supported on $C$. | It follows trivially from the definition that $f$ is locally constant on the complement of the [[Cantor set]] $C$: since the Cantor set is a set of Lebesgue measure zero, the derivative of $f$ vanishes almost everywhere. For the same reason, the [[Generalized derivative|distributional derivative]] of $f$ is a [[Singular measures|singular measure]] $\mu$ supported on $C$. | ||
− | The Cantor function is a prototype of [[Singular function]] in the sense of Lebesgue, | + | The Cantor function is a prototype of a [[Singular function|singular function]] in the sense of Lebesgue, cf. [[Lebesgue decomposition]]. The observation that $f$ is a function of bounded variation for which |
+ | \[ | ||
f (1)-f(0) \neq \int_0^a f'(t)\, dt | f (1)-f(0) \neq \int_0^a f'(t)\, dt | ||
\] | \] | ||
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|valign="top"|{{Ref|Ro}}|| H.L. Royden, "Real analysis" , Macmillan (1969). {{MR|0151555}} {{ZBL|0197.03501}} | |valign="top"|{{Ref|Ro}}|| H.L. Royden, "Real analysis" , Macmillan (1969). {{MR|0151555}} {{ZBL|0197.03501}} | ||
|- | |- | ||
− | |valign="top"|{{Ref|Vi}}|| A. Vitali,"Sulle funzioni integrali", Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., '''40''' 1905 pp. 1021-1034. | + | |valign="top"|{{Ref|Vi}}|| A. Vitali, "Sulle funzioni integrali", Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., '''40''' 1905 pp. 1021-1034. |
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Latest revision as of 11:26, 13 December 2014
2020 Mathematics Subject Classification: Primary: 26A45 Secondary: 28A15 [MSN][ZBL]
The Cantor ternary function (also called Devil's staircase and, rarely, Lebesgue's singular function) is a continuous monotone function $f$ mapping the interval $[0,1]$ onto itself, with the remarkable property that its derivative vanishes almost everywhere (recall that any monotone function is differentiable almost everywhere, see for instance Function of bounded variation).
![](/images/math/8/7/7/8775df11722a6c2878087f0c5d00e0db.png)
The function can be defined in the following way (see, for example, Exercise 46 in Chapter 2 of [Ro]). Given $x\in [0,1]$ consider its ternary expansion $\{a_i\}$, i.e. a choice of coefficients $a_i\in \{0,1,2\}$ such that \[ x = \sum_{i=1}^\infty \frac{a_i}{3^i}\, . \] Define $n(x)$ to be
- $\infty$ if none of the coefficients $a_i$ takes the value $1$
- the smallest integer $n$ such that $a_n=1$ otherwise.
Then \[ f(x) = \sum_{i=1}^{n(x)-1} \frac{a_i}{2^{i+1}} + \frac{1}{2^{n(x)}}. \] For alternative definitions see Example 1.67 of [AFP] and page 55 of [Co].
It follows trivially from the definition that $f$ is locally constant on the complement of the Cantor set $C$: since the Cantor set is a set of Lebesgue measure zero, the derivative of $f$ vanishes almost everywhere. For the same reason, the distributional derivative of $f$ is a singular measure $\mu$ supported on $C$.
The Cantor function is a prototype of a singular function in the sense of Lebesgue, cf. Lebesgue decomposition. The observation that $f$ is a function of bounded variation for which \[ f (1)-f(0) \neq \int_0^a f'(t)\, dt \] (where $f'$ denotes the classical pointwise derivative) was first made by Vitali in [Vi]. For this reason some authors use the terminology Cantor-Vitali function (see [AFP]).
References
[AFP] | L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001 |
[Co] | D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993. |
[Ro] | H.L. Royden, "Real analysis" , Macmillan (1969). MR0151555 Zbl 0197.03501 |
[Vi] | A. Vitali, "Sulle funzioni integrali", Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 40 1905 pp. 1021-1034. |
Cantor ternary function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cantor_ternary_function&oldid=27834