(4) A construction of a continuous nowhere differentiable function on R. We use [x] (resp. [x]) to denote the greatest i

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4 A Construction Of A Continuous Nowhere Differentiable Function On R We Use X Resp X To Denote The Greatest I 1 (136.05 KiB) Viewed 27 times

(4) A construction of a continuous nowhere differentiable function on R. We use [x] (resp. [x]) to denote the greatest integer less than or equal to x (resp. the least integer greater than or equal to x). (a) Let w: R→ R be given by w(x) = (c) For x (4m x = which resembles a wave. Show that w is a periodic function with period t = 2 and that for any interval (a, b) such that (a, b) nZ = Ø, we have |w(b) w(a)| b-a (b) Prove that there is a continuous function f: R→ R with the formula xx if [x] is even - [x]-xif [x] is odd n f(x) = Σ (²³) w(4¹ x) 4 n=0 R and m € Z+, the interval (4mx - 1/2,4mx + 1/2) has length 1. Thus, 1/2,4mx) or (4mx, 4mx + 1/2) does not contain an integer. Let 4-m 8m = ±² 2 with the sign chosen so that there are no integers between 4mx and 4m (x + 8m). Using part a, prove the following w(4" (x + 8m)) - w(4") 8m Sm := 4n 0 if 0 ≤ n ≤m if n > m (d) Using the previous part, give a lower bound for | f(x+8m) - f(x) | 8m In particular, show that sm→ ∞ as m → ∞. 1