University of Toronto – MAT237Y1 – LEC5201 Multivariable calculus Darboux’s construction of Riemann’s integral in one variable Jean-Baptiste Campesato January 21st, 2020 Contents 1 Historical comments 2 2 Darboux’s construction of Riemann’s integral 3 3 The 휀-criterion for integrability 5 4 Properties of Riemann’s integral 6 5 Some sufficient conditions for integrability 9 5.1 Monotonicity ......................................... 9 5.2 Continuity .......................................... 10 5.3 Discontinuity set has zero content ............................ 10 6 The MVT for Riemann’s integral 12 7 The Fundamental Theorem of Calculus 12 8 Riemann sums 14 A Supremum and Infimum 15 B Uniform continuity 17 C Riemann’s integrability criterion in terms of oscillation (Addendum from Feb 3) 20 D Lebesgue criterion for Riemann integrability (Addendum from Feb 3, extra-curricular, not part of MAT237) 23 2 Darboux’s construction of Riemann’s integral 1 Historical comments • Several methods allowing us to compute areas by finer approximations were already known in ancient Greece (Eudoxus’ method of exhaustion, Archimedes’ triangles…). • During the 17th century Gregory, Barrow, Newton on one side and Leibniz on the other side independently proved the FTC (where the integral was defined as the area under the curve for a continuous function on a segment line). • Cauchy (”Résumé des Leçons sur le calcul infinitésimal”, 1823) gave a first constructive definition of “an integral” in terms of Cauchy sums (which are left-Riemann sumsusing today’s terminology) but he restricted himself to continuous functions. • Riemann (”La possibilité de représenter une fonction par une série trigonométrique” 1854 but published in 1873) generalized Cauchy’s definition and removed the continuous assump- tion. The question at this time was to determine the largest class of functions for which we can compute integrals. He gave a non-constructible characterization of integrability in terms of “oscillation of a function” (this criterion has been superseded by Lebesgue’s criterion for Riemann integrability, in terms of discontinuity set, proved in the below cited paper). Riemann’s paper is very important because it allowed mathematicians to construct examples of continuous functions which are not differentiable. Indeed, if 푓 ∶ [푎, 푏] → ℝ is Riemann- 푥 integrable then 퐹 (푥) = ∫푎 푓(푡)푑푡 is uniformly continuous on [푎, 푏], but if we start with a non- continuous 푓, we may obtain a non-differentiable 퐹 ; hence such a 퐹 is continuous but not differentiable. Before that most mathematicians believed that continuity implies differentiability. • Darboux (”Mémoire sur les fonctions discontinues” 1875) gave a new definition equivalent to Riemann’s integral. This is the construction we are going to present in these notes. In this paper (§IX, pp. 109–110), among other results, Darboux also proved that a derivative may not be continuous (e.g. starting with 푓(푥) = 푥2 sin(1/푥) for 푥 ≠ 0 and 0 otherwise) but, nevertheless, the conclusion of the IVT always holds for a derivative (even if it is not contin- uous!). The latter result is usually called “Darboux’s theorem” and a function satisfying the IVT property while not being continuous (e.g. 푓 ′ for the above 푓) is usually called a “Dar- boux function”. He also constructed a function everywhere differentiable but whose derivative is not con- tinuous on the rational numbers: by Darboux’s theorem, such a function satisfies the IVT property while being discontinuous on any interval. • Lebesgue (”Leçons sur l’intégration et la recherche des fonctions primitives” 1904 after a CRAS note in 1901) gave an axiomatic definition of ”an” integral operator. This led himto introduce measure theory and Lebesgue’s integral (which is probably the common point of view of mathematicians nowadays). • There are other integration theories: • The Riemann–Stieltjes integral. It may be seen as a weighted version of Riemann’s in- tegral. It was introduced by Stieltjes in order to modelize mass distributions on the real line in 1894. • The Henstock–Kurzweil integral. It was first defined by Denjoy in 1912, but thereare several equivalent constructions (the nowadays common construction is due to Kurweil (1957) and was then developed by Henstock, it is similar to Riemann’s integral construc- tion with tagged partitions but it involves the notion of gauge). This integral is a little bit more difficult to construct than Riemann’s integral, but it admits some powerful results available in Lebesgue’s integral (e.g. monotone convergence theorem, dominated con- vergence theorem). Besides, the HK integral behaves well w.r.t. “improper integrals”. • … MAT237Y1 – LEC5201 – J.-B. Campesato 3 2 Darboux’s construction of Riemann’s integral Definition 1. A partition 푃 of the segment line [푎, 푏] consists in breaking [푎, 푏] into finitely many closed subintervals. We simply describe it by giving the boundaries of the subintervals: 푃 = {푎 = 푥0 < 푥1 < ⋯ < 푥푛 = 푏} Hence 푃 is a finite set of points of [푎, 푏] containing the endpoints 푎 and 푏. 푥 = 푎 푥 푥 푥 푥 0 1 2 3 푛−1 푥푛 = 푏 Definition 2. Let 푓 ∶ [푎, 푏] → ℝ be a bounded function and let 푃 = {푎 = 푥0 < 푥1 < ⋯ < 푥푛 = 푏} be a partition of [푎, 푏]. We define the upper Darboux sum of 푓 with respect to 푃 by 푛 푈 (푓) = (푥 − 푥 ) 푓 푃 ∑ 푘 푘−1 sup 푘=1 ( [푥푘−1,푥푘] ) and the lower Darboux sum of 푓 with respect to 푃 by 푛 퐿푃 (푓) = (푥푘 − 푥푘−1) inf 푓 ∑ ( [푥 ,푥 ] ) 푘=1 푘−1 푘 In the following figure, the upper Darboux sum is the area of the light grey and the darkgrey rectangles together whereas the lower Darboux sum is the area of the dark grey rectangles only. 푥0 푥1 푥2 푥3 푥4 Remark 3. Notice that the assumption “푓 is bounded” ensures that the Darboux sums are well- defined. Indeed, then the infimum and the supremum on the subintervals exist thanks totheLUB and GLB principles. Proposition 4. For any partition 푃 of [푎, 푏], we have 푈푃 (푓) ≥ 퐿푃 (푓). Proof. 푛 푈 (푓) = (푥 − 푥 ) 푓 푃 ∑ 푘 푘−1 sup 푘=1 ( [푥푘−1,푥푘] ) 푛 ≥ (푥푘 − 푥푘−1) inf 푓 since 푥푘 > 푥푘−1 and sup 푓 ≥ inf 푓 ∑ ( [푥 ,푥 ] ) 푘=1 푘−1 푘 = 퐿푃 (푓) ■ 4 Darboux’s construction of Riemann’s integral Definition 5. Let 푃 and 푄 be two partitions of [푎, 푏]. We say that 푄 is finer than 푃 if 푃 ⊂ 푄. 푃 푄 Proposition 6. If 푄 is finer than 푃 then 푈푄(푓) ≤ 푈푃 (푓) and 퐿푄(푓) ≥ 퐿푃 (푓) Proof. By induction, it is enough to see what happens if we break one subinterval into two subin- tervals. I am just doing it for the upper sum. Let 푐 ∈ (푥푘−1, 푥푘). Then (푥푘 − 푥푘−1) sup 푓 = (푥푘 − 푐 + 푐 − 푥푘−1) sup 푓 [푥푘−1,푥푘] [푥푘−1,푥푘] = (푐 − 푥푘−1) sup 푓 + (푥푘 − 푐) sup 푓 [푥푘−1,푥푘] [푥푘−1,푥푘] ≥ (푐 − 푥푘−1) sup 푓 + (푥푘 − 푐) sup 푓 [푥푘−1,푐] [푐,푥푘] ■ Proposition 7. For any partitions 푃 and 푄 of [푎, 푏], we have 퐿푃 (푓) ≤ 푈푄(푓). Proof. Indeed, set 푅 = 푃 ∪ 푄 then 푅 is finer than 푃 and finer than 푄, so 퐿푃 (푓) ≤ 퐿푅(푓) ≤ 푈푅(푓) ≤ 푈푄(푓) ■ Definition 8. Let 푓 ∶ [푎, 푏] → ℝ be a bounded function. We define the lower (Darboux) integral of 푓 by 푏 푓 = sup 퐿 (푓), ∀푃 partition of [a,b] ⨜ { 푃 } 푎 and the upper (Darboux) integral of 푓 by 푏 푓 = inf {푈푃 (푓), ∀푃 partition of [a,b]} ⨛푎 lower sums upper sums ⨜푏 ≤ 푏 푎 푓 ⨛푎 푓 finer partitions finer partitions Definition 9. Let 푓 ∶ [푎, 푏] → ℝ be a bounded function. ⨜푏 푏 We say that 푓 is integrable on [푎, 푏] if 푎 푓 = ⨛푎 푓. Then we denote this quantity by 푏 푓(푥)푑푥 ∫푎 MAT237Y1 – LEC5201 – J.-B. Campesato 5 3 The 휀-criterion for integrability Theorem 10. Let 푓 ∶ [푎, 푏] → ℝ be a bounded function. Then 푓 is integrable on [푎, 푏] if and only if ∀휀 > 0, ∃ a partition 푃 of [푎, 푏], 푈푃 (푓) − 퐿푃 (푓) < 휀 Remark 11. This criterion is not constructible: it doesn’t give the value of the integral! Proof. ⇒∶ We know that 푓 is integrable on [푎, 푏], i.e. 푏 푏 (1) 푓 = 푓 ⨜ 푎 ⨛푎 푏 푏 where 푓 = sup 퐿 (푓), ∀푃 partition of [a,b] and 푓 = inf 푈 (푓), ∀푃 partition of [a,b] . ⨜ { 푃 } { 푃 } 푎 ⨛푎 We want to prove: ∀휀 > 0, ∃ a partition 푃 of [푎, 푏], 푈푃 (푓) − 퐿푃 (푓) < 휀 Let 휀 > 0. 푏 푏 Then 푓 + 휀 is greater than 푓 which is the greatest lower bound of the upper Darboux sums. ⨛푎 2 ⨛푎 푏 Hence 푓 + 휀 is not an lower bound of the upper Darboux sums. ⨛푎 2 That means that there exists a partition 푃1 of [푎, 푏] such that 푏 휀 푈 (푓) < 푓 + 푃1 ⨛푎 2 푏 푏 Similarly ⨜ 푓 − 휀 is less than ⨜ 푓 which is the least upper bound of the lower Darboux sums. 푎 2 푎 푏 Hence ⨜ 푓 − 휀 is not an upper bound of the lower Darboux sums. 푎 2 That means that there exists a partition 푃2 of [푎, 푏] such that 푏 휀 퐿 (푓) > 푓 − 푃2 ⨜ 푎 2 Let 푃 = 푃1 ∪ 푃2. Then 푃 is finer than 푃1, hence 푏 휀 (2) 푈 (푓) ≤ 푈 (푓) < 푓 + 푃 푃1 ⨛푎 2 and similarly 푃 is finer than 푃2, hence 푏 휀 (3) 퐿 (푓) ≥ 퐿 (푓) > 푓 − 푃 푃2 ⨜ 푎 2 We derive from (2) and (3) that 푏 푏 휀 휀 푈 (푓) − 퐿 (푓) < 푓 + − 푓 + 푃 푃 ⨜ ⨛푎 2 푎 2 Using (1), we obtain that the RHS of the above inequality is 휀.