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

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4 A Construction Of A Continuous Nowhere Differentiable Function On R We Use Z Resp X To Denote The Greatest I 1 (61.57 KiB) Viewed 21 times

(4) A construction of a continuous nowhere differentiable function on R. We use [z] (resp. [x]) to denote the greatest integer less than or equal to z (resp. the least integer greater than or equal to x). (a) Let w: R → R be given by w(x) = = I- [x] ev if [x] is odd 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 = 0, we have |w(b) w(a)| b-a (b) Prove that there is a continuous function f: R→ R with the formula DO 72 f(x) = (³) * w (4¹ x) n=0 (c) For ER and m € Z+, the interval (4"-1/2,4x + 1/2) has length 1. Thus, (4 x 1/2, 4x) or (4mx, 4x+1/2) does not contain an integer. Let 8m := 8m = ±477 4-m 2 with the sign chosen so that there are no integers between 4 and 4" (x + 8m). Using part a, prove the following |w(4¹(x+8m)) - w(4")| 4n if 0 ≤ n ≤m if n > m 8m (d) Using the previous part, give a lower bound for | ƒ(x + 8m) - f(x) 8m ) = f(x) || In particular, show that 8m →∞ as m → ∞o. 1 MATH 424 HOMEWORK 2 DUE JULY 11, 2022 (e) Note that (x + 8m)m>0 is a sequence that converges to z. Conclude that f is not differentiable at z.