1.6. Differentiation theorems 129 The diﬃculty here is that a continuous function can still contain a large amount of oscillation, which can lead to breakdown of differentiability. How- ever, if one can somehow limit the amount of oscillation present, then one can often recover a fair bit of differentiability. For instance, we have Theorem 1.6.25 (Monotone differentiation theorem). Any function F : R → R which is monotone (either monotone non-decreasing or monotone non-increasing) is differentiable almost everywhere. Exercise 1.6.29. Show that every monotone function is measurable. To prove this theorem, we just treat the case when F is monotone non- decreasing, as the non-increasing case is similar (and can be deduced from the non-decreasing case by replacing F with −F ). We also first focus on the case when F is continuous, as this allows us to use the rising sun lemma. To understand the differentiability of F , we introduce the four Dini derivatives of F at x: (i) The upper right derivative D+F (x) := lim suph→0+ F (x+h)−F (x) h (ii) The lower right derivative D+F (x) := lim infh→0+ F (x+h)−F (x) h (iii) The upper left derivative D−F (x) := lim suph→0− F (x+h)−F (x) h (iv) The lower right derivative D−F (x) := lim infh→0− F (x+h)−F (x) h . Regardless of whether F is differentiable or not (or even whether F is con- tinuous or not), the four Dini derivatives always exist and take values in the extended real line [−∞, ∞]. (If F is only defined on an interval [a, b], rather than on the endpoints, then some of the Dini derivatives may not exist at the endpoints, but this is a measure zero set and will not impact our analysis.) Exercise 1.6.30. If F is monotone, show that the four Dini derivatives of F are measurable. (Hint: The main diﬃculty is to reformulate the derivatives so that h ranges over a countable set rather than an uncountable one.) A function F is differentiable at x precisely when the four derivatives are equal and finite: (1.29) D+F (x) = D+F (x) = D−F (x) = D−F (x) ∈ (−∞, +∞). We also have the trivial inequalities D+F (x) ≤ D+F (x) D−F (x) ≤ D−F (x). If F is non-decreasing, all these quantities are non-negative, thus 0 ≤ D+F (x) ≤ D+F (x) 0 ≤ D−F (x) ≤ D−F (x).

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