Difference between revisions of "Devil's staircase"

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[https://www.youtube.com/watch?v=dQXVn7pFsVI The Devil's Staircase | Infinite Series (19 May 2017)]<br />
 
[https://www.youtube.com/watch?v=dQXVn7pFsVI The Devil's Staircase | Infinite Series (19 May 2017)]<br />
 
[https://www.youtube.com/watch?v=Hv3fxroMt1s Devil's Staircase (19 February 2017)]<br />
 
[https://www.youtube.com/watch?v=Hv3fxroMt1s Devil's Staircase (19 February 2017)]<br />
 +
[https://www.youtube.com/watch?v=TFwiU3W_HoI Intro Real Analysis, Lec 15, Uniform Continuity, Monotone Functions, Devil's Staircase, Derivatives (5 October 2016)]<br />
 
[https://www.youtube.com/watch?v=sjfgim3hrno Cantor's staircase (25 November 2014)]<br />
 
[https://www.youtube.com/watch?v=sjfgim3hrno Cantor's staircase (25 November 2014)]<br />
  

Revision as of 16:03, 10 July 2017

The Devil's staircase (also known as the Cantor function) is a function $c \colon [0,1] \rightarrow [0,1]$ can be expressed by the following rules:

  1. Write $x$ in base-3.
  2. If that representation of $x$ contains a $1$, replace every digit after the first $1$ with $0$'s.
  3. Replace all $2$'s with $1$'s.
  4. The resulting expansion defines $c(x)$.

Properties

Devil's staircase is continuous
Devil's staircase is not absolutely continuous

Videos

The Devil's Staircase | Infinite Series (19 May 2017)
Devil's Staircase (19 February 2017)
Intro Real Analysis, Lec 15, Uniform Continuity, Monotone Functions, Devil's Staircase, Derivatives (5 October 2016)
Cantor's staircase (25 November 2014)

References

Cantor function