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CantorStaircase[x]

gives the Cantor staircase function .

Details
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Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Function Properties  
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CantorStaircase

CantorStaircase[x]

gives the Cantor staircase function .

Details

  • The Cantor staircase function is also known as Cantor ternary function or Cantor function.
  • Mathematical function, suitable for both symbolic and numeric manipulation.
  • For , the Cantor function equals TemplateBox[{x}, CantorStaircase]=sum_i2^(-n_i).
  • For certain arguments, CantorStaircase automatically evaluates to exact values.
  • CantorStaircase can be evaluated to arbitrary numerical precision.
  • CantorStaircase automatically threads over lists. »

Examples

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Basic Examples  (2)

Evaluate at a point in the Cantor set:

Wolfram Language code: CantorStaircase[1 / 3]

Plot CantorStaircase over the unit interval:

Wolfram Language code: Plot[CantorStaircase[x], {x, 0, 1}]

Scope  (13)

Numerical Evaluation  (5)

Evaluate numerically:

Wolfram Language code: CantorStaircase[0.76]

Evaluate with integer inputs:

Wolfram Language code: CantorStaircase[0]

Evaluate at rational numbers:

Wolfram Language code: CantorStaircase[15 / 17]

Evaluate at high precision:

Wolfram Language code: N[CantorStaircase[1 / Sqrt[2]], 20]

Compute the elementwise values of an array using automatic threading:

Wolfram Language code: CantorStaircase[{{1 / 2, 0}, {0, 1 / 2}}]

Or compute the matrix CantorStaircase function using MatrixFunction:

Wolfram Language code: MatrixFunction[CantorStaircase, {{1 / 2, 0}, {0, 1 / 2}}]//FullSimplify

Function Properties  (8)

CantorStaircase is defined for all real numbers:

Wolfram Language code: FunctionDomain[CantorStaircase[x], x]

Its domain is restricted to real inputs:

Wolfram Language code: FunctionDomain[CantorStaircase[x], x, Complexes]

The range of CantorStaircase:

Wolfram Language code: FunctionRange[CantorStaircase[x], x, y]

Since its range is bounded, it is not surjective:

Wolfram Language code: FunctionSurjective[CantorStaircase[x], x]
Wolfram Language code: Plot[{CantorStaircase[x], 2}, {x, -2, 2}]

CantorStaircase is not injective:

Wolfram Language code: FunctionInjective[CantorStaircase[x], x]
Wolfram Language code: Plot[{CantorStaircase[x], 1}, {x, -2, 2}]

CantorStaircase is continuous:

Wolfram Language code: FunctionContinuous[CantorStaircase[x], x]

CantorStaircase is nondecreasing:

Wolfram Language code: FunctionMonotonicity[CantorStaircase[x], x]

CantorStaircase is non-negative:

Wolfram Language code: FunctionSign[CantorStaircase[x], x]

CantorStaircase is neither convex nor concave:

Wolfram Language code: FunctionConvexity[CantorStaircase[x], x]

TraditionalForm formatting:

Wolfram Language code: CantorStaircase[x]//TraditionalForm
Wolfram Research (2014), CantorStaircase, Wolfram Language function, https://reference.wolfram.com/language/ref/CantorStaircase.html.

Text

Wolfram Research (2014), CantorStaircase, Wolfram Language function, https://reference.wolfram.com/language/ref/CantorStaircase.html.

CMS

Wolfram Language. 2014. "CantorStaircase." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CantorStaircase.html.

APA

Wolfram Language. (2014). CantorStaircase. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CantorStaircase.html

BibTeX

@misc{reference.wolfram_2026_cantorstaircase, author="Wolfram Research", title="{CantorStaircase}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/CantorStaircase.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_cantorstaircase, organization={Wolfram Research}, title={CantorStaircase}, year={2014}, url={https://reference.wolfram.com/language/ref/CantorStaircase.html}, note=[Accessed: 13-June-2026]}