1.6. Differentiation theorems 147 Remark 1.6.42. The above proposition is yet another illustration of how the property of everywhere differentiability is significantly better than that of almost everywhere differentiability. In practice, though, the above propo- sition is not as useful as one might initially think, because there are very few methods that establish the everywhere differentiability of a function that do not also establish continuous differentiability (or at least Riemann integra- bility of the derivative), at which point one could just use Theorem 1.6.7 instead. Exercise 1.6.53. Let F : [−1, 1] → R be the function defined by setting F (x) := x2 sin( 1 x3 ) when x is non-zero, and F (0) := 0. Show that F is everywhere differentiable, but the deriative F is not absolutely integrable, and so the second fundamental theorem of calculus does not apply in this case (at least if we interpret [a,b] F (x) dx using the absolutely convergent Lebesgue integral). See, however, the next exercise. Exercise 1.6.54 (Henstock-Kurzweil integral). Let [a, b] be a compact in- terval of positive length. We say that a function f : [a, b] → R is Henstock- Kurzweil integrable with integral L ∈ R if for every ε 0 there exists a gauge function δ : [a, b] → (0, +∞) such that one has | k j=1 f(tj ∗)(tj − tj−1) − L| ≤ ε whenever k ≥ 1 and a = t0 t1 . . . tk = b and t1,...,tk ∗ ∗ are such that tj ∗ ∈ [tj−1,tj] and |tj − tj−1| ≤ δ(tj ∗) for every 1 ≤ j ≤ k. When this occurs, we call L the Henstock-Kurzweil integral of f and write it as [a,b] f(x) dx. (i) Show that if a function is Henstock-Kurzweil integrable, it has a unique Henstock-Kurzweil integral. (Hint: Use Cousin’s theorem.) (ii) Show that if a function is Riemann integrable, then it is Henstock- Kurzweil integrable, and the Henstock-Kurzweil integral [a,b] f(x) dx is equal to the Riemann integral b a f(x) dx. (iii) Show that if a function f : [a, b] → R is everywhere defined, ev- erywhere finite, and is absolutely integrable, then it is Henstock- Kurzweil integrable, and the Henstock-Kurzweil integral [a,b] f(x) dx is equal to the Lebesgue integral [a,b] f(x) dx. (Hint: This is a variant of the proof of Theorem 1.6.40 or Proposition 1.6.41.) (iv) Show that if F : [a, b] → R is everywhere differentiable, then F is Henstock-Kurzweil integrable, and the Henstock-Kurzweil integral [a,b] F (x) dx is equal to F (b) − F (a). (Hint: This is a variant of the proof of Theorem 1.6.40 or Proposition 1.6.41.)

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