Answer Happy

(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 r). (a) Let w: R→ R be given by w(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)^Z = Ø, we have |w(b) — w(a)| = b-a (b) Prove that there is a continuous function ƒ: R → R with the formula [x−[r]_if [x] is even [x] − x if [x] is odd f(x) = -2 (;))" = Σ w (4x) (c) For r ER and m € Z₁, the interval (4mx − 1/2,4mx + 1/2) has length 1. Thus, (4mx 1/2,4mx) or (4mx, 4mx+1/2) does not contain an integer. Let 4-m 8m = ±- 2 with the sign chosen so that there are no integers between 4 and 4m (x + 8m). Using part a, prove the following Sm |w(4″ (x+8m)) − w (4¹) 8m ²1-{8 = 0 4n if 0 ≤ n ≤m if n > m (d) Using the previous part, give a lower bound for |ƒ(x+8m) − f (x) 8m In particular, show that sm → ∞ as m → ∞.