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Unformatted text preview: The Devils Staircase Recall the usual construction of the Cantor set: C = [0 , 1] , C 1 = [0 , 1 3 ] [ 2 3 , 1] , and in general C n is a disjoint union of 2 n closed intervals, each of length 3- n , constructed from C n- 1 by deleting the open-middle-third of each of the 2 n- 1 intervals constituting C n- 1 . Then the total length of the intervals in C n is ( 2 3 ) n . The Cantor set C is the intersection C = T n C n . Set g n = ( 3 2 ) n 1 C n ; that is, g n ( x ) = ( ( 3 2 ) n , x C n , x / C n . The function g n is discontinuous only at the 2 n +1 points at the boundaries of the intervals making up C n . This is a finite set, and so g n is Riemann integrable. So we may define f n : [0 , 1] R as follows: f n ( x ) = Z x g n ( t ) dt. From the Fundamental Theorem of Calculus, we know that the functions f n are Lipschitz continuous. Note also that f n (0) = 0 , while f n (1) = R 1 g n ( t ) dt . This integral can be calculated as Z 1 g n ( t ) dt = ( 3 2 ) n R 1 1 C n ( t ) dt = ( 3 2 ) n length ( C n ) = 1 . So f n is a continuous function with f n (0) = 0 and f n (1) = 1 for each n . Notice that f n is constant on [0 , 1]- C n , and is linear with slope ( 2 3 ) n on the intervals making up C n . So f n is monotone increasing. Here is the graph ofis monotone increasing....
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This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.
- Fall '10