1.6. Differentiation theorems 141 (iii) Show that for each n = 0, 1,..., one has |Fn+1(x) Fn(x)| 2−n for each x [0, 1]. Conclude that the Fn converge uniformly to a limit F : [0, 1] R. This limit is known as the Cantor function. (iv) Show that the Cantor function F is continuous and monotone non- decreasing, with F (0) = 0 and F (1) = 1. (v) Show that if x [0, 1] lies outside the middle thirds Cantor set (Exercise 1.2.9), then F is constant in a neighbourhood of x, and in particular, F (x) = 0. Conclude that [0,1] F (x) dx = 0 = 1 = F (1) F (0), so that the second fundamental theorem of calculus fails for this function. (vi) Show that F ( ∑∞ n=1 an3−n)= ∑∞ n=1 an 2 2−n for any digits a1,a2,...∈ {0, 2}. Thus the Cantor function, in some sense, converts base three expansions to base two expansions. (vii) Let I = [ ∑n i=1 ai 3i , ∑n i=1 ai 3i + 1 3n ] be one of the intervals used in the nth cover In of C (see Exercise 1.2.9), thus n 0 and a1,...,an {0, 2}. Show that I is an interval of length 3−n, but F (I) is an interval of length 2−n. (viii) Show that F is not differentiable at any element of the Cantor set C. Remark 1.6.38. This example shows that the classical derivative F (x) := limh→0 h=0 F (x+h)−F (x) h of a function has some defects it cannot “see” some of the variation of a continuous monotone function such as the Cantor func- tion. In §1.13 of An epsilon of room, Vol. I, this will be rectified by intro- ducing the concept of the weak derivative of a function, which despite the name, is more able than the strong derivative to detect this type of singular variation behaviour. (We will also encounter in Section 1.7.3 the Lebesgue- Stieltjes integral, which is another (closely related) way to capture all of the variation of a monotone function, and which is related to the classical derivative via the Lebesgue-Radon-Nikodym theorem see §1.2 of An epsilon of room, Vol. I.) In view of this counterexample, we see that we need to add an additional hypothesis to the continuous monotone non-increasing function F before we can recover the second fundamental theorem. One such hypothesis is absolute continuity. To motivate this definition, let us recall two existing definitions: (i) A function F : R R is continuous if, for every ε 0 and x0 R, there exists a δ 0 such that |F (b) F (a)| ε whenever (a, b) is an interval of length at most δ that contains x0.
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