|
To view the graphing board, you need
Flash Player 6 or higher. Click
below to download the free player from the Macromedia site.
Download Flash Player.
To graph the function, press the NEXT
STEP button repeatedly. Experiment with zooming in and out to get the feel for the graph.
COMMENTS:
As you know, the Cantor function, f, is a continuous, increasing function from [0,1] onto [0,1].
The function is constant on each of the subintervals [1/3,2/3],
[1/9,2/9], [7/9,8/9]... whose interiors form the complement of the Cantor set. Of course, we
cannot really draw the graph of f. The grapher above can display up to ten steps. No wonder,
we are dealing with indivisible units called pixels. At the n-th step, the length of each small subinterval
is 1/(3n), so it shrinks very fast. Below is a question for you to ponder.
QUESTION:
Suppose we can draw the graph of f on each of the subinervals [1/3,2/3], [1/9,2/9], [7/9, 8/9], and so on,
have we drawn the graph of f? In other words, does the union of the closed subintevals, whose interiors form
the complement of the Cantor set, equal [0,1]?
|