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Cantor function (Definition)

The Cantor function is a canonical example of a singular function. It is based on the Cantor set, and is usually defined as follows. Let $ x$ be a real number in $ [0,1]$ with the ternary expansion $ 0.a_1 a_2 a_3 \ldots$, then let $ N$ be $ \infty$ if no $ a_n = 1$ and otherwise let $ N$ be the smallest value such that $ a_n = 1$. Next let $ b_n = \frac{1}{2}a_n$ for all $ n < N$ and let $ b_N = 1$. We define the Cantor function (or the Cantor ternary function) as

$\displaystyle f(x) = \sum_{n=1}^N \frac{b_n}{2^n}.$    

This function can be easily checked to be continuous and monotonic on $ [0,1]$ and also $ f'(x) = 0$ almost everywhere (it is constant on the complement of the Cantor set), with $ f(0) = 0$ and $ f(1) = 1$. Another interesting fact about this function is that the arclength of the graph is 2, hence the calculus arclength formula does not work in this case.

\includegraphics[width=5.19in]{cantorfunction.eps}

Figure 1: Graph of the cantor function using 20 iterations.

This function, and functions similar to it are frequently called the Devil's staircase. Such functions sometimes occur naturally in various areas of mathematics and mathematical physics and are not just a pathological oddity.

Bibliography

1
H. L. Royden. Real Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1988



"Cantor function" is owned by jirka.
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See Also: Cantor set, singular function

Other names:  Cantor ternary function, Cantor-Lebesgue function, Devil's staircase
Also defines:  Cantor function, Cantor ternary function
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Cross-references: pathological, areas, similar, iterations, formula, Calculus, graph, complement, almost everywhere, monotonic, continuous, function, real number, Cantor set, singular function, canonical
There are 2 references to this entry.

This is version 6 of Cantor function, born on 2004-02-08, modified 2007-12-12.
Object id is 5554, canonical name is CantorFunction.
Accessed 19451 times total.

Classification:
AMS MSC: 26A30 (Real functions :: Functions of one variable :: Singular functions, Cantor functions, functions with other special properties)
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Devil's staircase by archibal on 2004-04-06 21:01:49
A nice article on Cantor's function (the Devil's staircase) finishes with a tanatlizing comment: Such functions turn up in many areas of mathematics and are not just pathological examples.

Nice to say that, but it really provokes the question: where do they turn up?

The only example I can think of off-hand is of locally constant functions on $p$-adic fields, and that's a bit iffy: the base field is pretty different from $\mathbb{R}$.

I suppose they provide a counterexample for the simplest interpretation of the Fundamental Theorem of Calculus, but that hardly helps them qualify as non-pathological.
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