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 x ). (a) Let w:R→R be given by w(x)={x−⌊x⌋⌈x⌉−x​ if ⌊x⌋ is even  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)∩Z=∅, we have ∣w(b)−w(a)∣=b−a (b) Prove that there is a continuous function f:R→R with the formula f(x)=n=0∑∞​(43​)nw(4nx) (c) For x∈R and m∈Z+​, 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 δm​=±24−m​ with the sign chosen so that there are no integers between 4mx and 4m(x+δm​). Using part a, prove the following ∣∣​δm​w(4n(x+δm​))−w(4n)​∣∣​={4n0​ if 0≤n≤m if n>m​ (d) Using the previous part, give a lower bound for sm​:=∣∣​δm​f(x+δm​)−f(x)​∣∣​ In particular, show that sm​→∞ as m→∞.