Cantor Function

The Cantor function is a function that is continuous, differentiable, increasing, non-constant, and the derivative is zero everywhere except at a set with length zero. It is the most difficult function in our repertoire and can be found, for example, in Kolmogorov and Fomin.

Recall the definition of the Cantor set: Let

= [ 1/3, 2/3]

be the middle third of the interval [0, 1]. Let

= [ 1/9, 2/9], = [ 7/9, 8/9 ]

be the middle thirds of the intervals remaining after deleting

from [0, 1]. Let

= [2/27, 3/27], = [7/27, 8/27],
= [19/27, 20/27], = [ 25/27, 26,27]

be the middle thirds of the intervals remaining after deleting

, and

from [0, 1]. Continue in this fashion so that at the n-th stage we have the intervals

, ..., , ...,

Then the complement of the union of all these intervals

is the Cantor set without the endpoints. Now define the following function

if t is in the interval

Then, for example, we have that

if 1/3 t 2/3

Then F(t) is defined everywhere in [0, 1] except at the Cantor set minus the end points 0, 1, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, ... If t is a number where F is not defined, then there exists an increasing sequence { } of these endpoints converging to t, and a decreasing sequence { ' } of these endpoints converging to t. Since F is defined at those endpoints and ', we define

F(t) = F() = F(')

Now we have defined completely the Cantor function. It has the following properties:

F is defined everywhere in the interval [0, 1]
F is not constant
F is increasing
F is continuous in the interval [0, 1]
F is differentiable in the interval [0, 1]
F' is zero at every interior point of the intervals

In particular, F' is zero at points of total length 1 in the interval [0, 1], yet F it is not constant.

Proof

Some of these properties are obvious, and some require more thought. In particular, why does the above limit of endpoints exist ? That is a crucial point, because we used this limit to extend the function to the whole interval [0, 1]. For details and hints about the Cantor Function, please consult Kolmogorov and Fomin, p 334 ff.

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