1.6. Differentiation theorems 143 (ii) Show that the Cantor function does not have this property. For absolutely continuous functions, we can recover the second funda- mental theorem of calculus: Theorem 1.6.40 (Second fundamental theorem for absolutely continuous functions). Let F : [a, b] R be absolutely continuous. Then [a,b] F (x) dx = F (b) F (a). Proof. Our main tool here will be Cousin’s theorem (Exercise 1.6.23). By Exercise 1.6.44, F is absolutely integrable. By Exercise 1.5.10, F is thus uniformly integrable. Now let ε 0. By Exercise 1.5.13, we can find κ 0 such that U |F (x)| dx ε whenever U [a, b] is a measurable set of measure at most κ. (Here we adopt the convention that F vanishes outside of [a, b].) By making κ small enough, we may also assume from absolute continuity that ∑n j=1 |F (bj) F (aj)| ε whenever (a1,b1),..., (an,bn) is a finite collection of disjoint intervals of total length ∑n j=1 bj aj at most κ. Let E [a, b] be the set of points x where F is not differentiable, together with the endpoints a, b, as well as the points where x is not a Lebesgue point of F thus E is a null set. By outer regularity (or the definition of outer measure) we can find an open set U containing E of measure m(U) κ. In particular, U |F (x)| dx ε. Now define a gauge function δ : [a, b] (0, +∞) as follows. (i) If x E, we define δ(x) 0 to be a small enough number such that the open interval (x δ(x),x + δ(x)) lies in U. (ii) If x E, then F is differentiable at x and x is a Lebesgue point of F . We let δ(x) 0 be a small enough number such that |F (y)−F (x)−(y−x)F (x)| ε|y−x| holds whenever |y−x| δ(x), and such that | 1 |I| I F (y) dy −F (x)| ε whenever I is an interval containing x of length at most δ(x) such a δ(x) exists by the defi- nition of differentiability, and of Lebesgue point. We rewrite these properties using big-O notation17 as F (y) F (x) = (y x)F (x) + O(ε|y x|) and I F (y) dy = |I|F (x) + O(ε|I|). Applying Cousin’s theorem, we can find a partition a = t0 t1 . . . tk = b with k 1, together with real numbers tj [tj−1,tj] for each 1 j k and tj tj−1 δ(tj ∗). 17In this notation, we use O(X) to denote a quantity Y whose magnitude |Y | is at most CX for some absolute constant C. This notation is convenient for managing error terms when it is not important to keep track of the exact value of constants such as C, due to such rules as O(X) + O(X) = O(X).
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