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Devil's-Staircase

Devil's-Staircase - The Devils Staircase Recall the usual...

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The Devil’s Staircase Recall the usual construction of the Cantor set: C 0 = [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 0 C n . Set g n = ( 3 2 ) n C n ; that is, g n ( x ) = ( ( 3 2 ) n , x C n 0 , 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 0 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 0 g n ( t ) dt . This integral can be calculated as Z 1 0 g n ( t ) dt = ( 3 2 ) n R 1 0 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 of f 2 .
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