Kenneth Kuttler

178 CHAPTER 7. THE INTEGRAL

Proof: The uniform convergence implies f is also continuous. See Theorem 4.9.3.Therefore,

∫ ba f dx exists. Using the triangle inequality and definition of ∥·∥ described ear-

lier in conjunction with this theorem,∣∣∣∣∫ b

af (x)dx−

∫ b

afn (x)dx

∣∣∣∣= ∣∣∣∣∫ b

a( f (x)− fn (x))dx

∣∣∣∣≤∫ b

a| f (x)− fn (x)|dx ≤

∫ b

a∥ f − fn∥dx ≤ ∥ f − fn∥(b−a)

which is given to converge to 0 as n → ∞.

7.3 The Riemann Darboux Integral∗

In the 1850’s Riemann gave a completely satisfactory description of the integral. The oneCauchy gave had some problems. I will present Darboux’s version of this integral and showthat it is the same as the earlier one for continuous and piecewise continuous functions. Iwill also present the fundamental theorem of calculus from this integral. In this section,[a,b] will represent the usual notion of an interval in which a < b.

Definition 7.3.1 For f a bounded function, and P = {x0,x1, ...,xn} ⊆ [a,b] where,a = x0 < · · ·< xn = b, let

Mi ( f )≡ sup{ f (x) : x ∈ [xi−1,xi]} , mi ( f )≡ inf{ f (x) : x ∈ [xi−1,xi]} .

Then upper sums, U ( f ,P) and lower sums L( f ,P) are defined as

U ( f ,P)≡n

∑i=1

Mi ( f )(xi − xi−1) ,L( f ,P)≡n

∑i=1

mi ( f )(xi − xi−1)

This collection of points is called a partition of [a,b].

What happens when you add in more points in a partition? In this example a singleadditional point, labeled z has been added in.

y = f (x)

x0 x1 x2 x3z

x0 x1 x2 x3

Note how the lower sum got larger by the amount of the area in the shaded rectangleand the upper sum got smaller by the amount in the other shaded rectangle. In general thisis the way it works and this is shown in the following lemma.

Lemma 7.3.2 If P ⊆ Q then

U ( f ,Q)≤U ( f ,P) , and L( f ,P)≤ L( f ,Q) .

178 CHAPTER 7. THE INTEGRALProof: The uniform convergence implies f is also continuous. See Theorem 4.9.3.Therefore, p? fdx exists. Using the triangle inequality and definition of ||-|| described ear-lier in conjunction with this theorem,[roe- [tora [re -meyarb b< fit) —filaes [It fille < IL fal Oa)which is given to converge to0 asn > 0, Jj7.3 The Riemann Darboux Integral*In the 1850’s Riemann gave a completely satisfactory description of the integral. The oneCauchy gave had some problems. I will present Darboux’s version of this integral and showthat it is the same as the earlier one for continuous and piecewise continuous functions. Iwill also present the fundamental theorem of calculus from this integral. In this section,[a,b] will represent the usual notion of an interval in which a < b.Definition 7.3.1 For f a bounded function, and P = {xo,x,...,%n} C [a,b] where,aA=XxXo <+++ <X,_, =), letMi (f) = sup {f (x) : x € [xi-1,x;]}, mi (f) = inf {f (*) : x © [i_-1,xi]}.Then upper sums, U (f,P) and lower sums L(f,P) are defined asIM:M:U(f,P) Mi(f) (xi —xi-1) L(F,P) = ) mi (f) i — 1-1)i=l 1This collection of points is called a partition of |a, b}.What happens when you add in more points in a partition? In this example a singleadditional point, labeled z has been added in.P= f(x) =XO xX] x2 z X3 XO xX] x2 z X3Note how the lower sum got larger by the amount of the area in the shaded rectangleand the upper sum got smaller by the amount in the other shaded rectangle. In general thisis the way it works and this is shown in the following lemma.Lemma 7.3.2 If P C Q thenU(f,Q) <U(f,P), and L(f,P) <L(f,Q).