216 CHAPTER 9. INTEGRATION

Proposition 9.10.2 With the above definition of length, L =∫ b

a |γ ′ (t)|dt.

Proof: Whenever considering P, one of these ordered partitions, there is no loss ofgenerality in assuming that the intermediate points z1,z2, ...,zn are in P because L(P) onlygets larger when points are added in to P. I will tacitly assume this in all that follows. Letf : [zk−1,zk]× [zk−1,zk]

f (s, t)≡

{(γ(t)−γ(s))−γ ′(s)(t−s)

t−s if t ̸= s0 if t = s

Then f is uniformly continuous due to continuity of γ ′ and compactness. Therefore thereexists δ k > 0 such that if |t− s|< δ k, then |(γ (t)− γ (s))− γ ′ (s)(t− s)|< ε

b−a |t− s|. Nowlet ∥P∥ < δ ≡ min{δ k,k = 1,2, ...,n} and always P includes the zk. Then by the triangleinequality, for such P,∣∣γ ′ (tk−1)

∣∣ |tk− tk−1|−ε

b−a|tk− tk−1| ≤ |γ (tk)− γ (tk−1)|

≤∣∣γ ′ (tk−1)

∣∣ |tk− tk−1|+ε

b−a|tk− tk−1| (9.12)

Thus, for ∥P∥ < δ , L(|γ ′| ,P)− ε ≤ L(P). Recall also the upper sums get smaller whenpoints are added and lower sums get larger. Therefore, there exists P with ∥P∥ < δ andU (|γ ′| ,P)−L(|γ ′| ,P)< ε . In particular, from the above inequality, L(P)≤ ∑P |γ ′|+ ε ≤∫ b

a |γ ′|dx+2ε so L≤∫ b

a |γ ′|dx+2ε . Thus, there exists possibly another P, with the aboveholding and also L− ε < L(P)≤ L. Then, from 9.12,∫ b

a

∣∣γ ′∣∣dx−2ε ≤ L(∣∣γ ′∣∣ ,P)− ε ≤ L(P)≤ L≤

∫ b

a

∣∣γ ′∣∣dx+2ε

and so L−∫ b

a |γ ′|dx ∈ [−2ε,2ε] . Since ε is arbitrary, it follows that L =∫ b

a |γ ′|dx.There are exactly two directions of motion over γ∗. In tracing out γ∗, one can either let

t go from a to b or from b to a and these are the only possibilities if γ is to be one to one.Indeed, if γ̂ maps the interval to γ∗ and is continuous and one to one, then γ̂

−1 ◦ γ is eitherstrictly increasing or strictly decreasing by Lemma 6.4.3. Increasing means same directionand decreasing, the opposite direction.

9.11 Exercises1. In the chapter, upper and lower sums were considered. Suppose g is an increasing

function and you are considering upper and lower sums for approximating∫ b

a f dg.Show that when you add in a point to the partition, the upper sum which results is nolarger but the lower sum is no smaller.

2. Let f (x) = 1+ x2 for x ∈ [−1,3] and let P ={−1,− 1

3 ,0,12 ,1,2

}. Find U ( f ,P) and

L( f ,P) for F (x) = x and for F (x) = x3.

3. Let P ={

1,1 14 ,1

12 ,1

34 ,2}

and F (x) = x. Find upper and lower sums for the functionf (x) = 1

x using this partition. What does this tell you about ln(2)?

4. If f ∈ R([a,b] ,F) with F (x) = x and f is changed at finitely many points, show thenew function is also in R([a,b] ,F) . Is this still true for the general case where F isonly assumed to be an increasing function? Explain.