180 CHAPTER 7. THE INTEGRAL

Definition 7.3.6 For f a bounded function on [a,b] ,

I ≡ inf{U ( f ,P) where P is a partition},I ≡ sup{L( f ,P) where P is a partition}.

Then f is integrable if Ī = I and the Darboux integral is the common value of these. I willcall I the lower integral and Ī the upper integral. Thus the function is integrable exactlywhen there is no gap between the upper and lower integrals.

Proposition 7.3.7 A bounded function f defined on [a,b] is integrable if and only if foreach ε > 0 there exists a partition P such that U ( f ,P)−L( f ,P)< ε .

Proof: ⇒ In this case, the upper and lower integrals are equal and so I +ε/3 > Ī,Ī−ε/3 < I . Thus, there is a partition P such that I +ε/3 >U ( f ,P) and Ī−ε/3 < L( f ,P).Therefore,

0 ≤U ( f ,P)−L( f ,P)≤ I + ε/3− (Ī − ε/3)< ε

⇐ If there is some P such that U ( f ,P)− L( f ,P) < ε, then 0 ≤ Ī − I ≤ U ( f ,P)−L( f ,P)< ε and so, since ε is arbitrary, Ī − I = 0. There is no gap between the upper andlower integrals.

Proposition 7.3.8 If f is either increasing or decreasing on [a,b] , then f is integrable.

Proof: Suppose first that f is decreasing. There is no space between Ī and I because ifa = x0 < x1 < · · ·< xn = b where these points in the partition are equally spaced, then

Ī − I ≤n

∑k=1

f (xk−1)(xk − xk−1)−n

∑k=1

f (xk)(xk − xk−1)

=n

∑k=1

( f (xk−1)− f (xk))b−a

n= ( f (a)− f (b))

b−an

Since n is arbitrary, it must be that Ī− I = 0. It is exactly similar for f increasing. You justtake the upper sum by using the value of f at the right end of the interval and the lowersum by taking the value of f at the left end of the interval.

Corollary 7.3.9 Suppose [a,b] is an interval and f is a bounded real valued functiondefined on this interval and that there is a partition a = z0 < z1 < · · ·< zn = b such that fis either increasing or decreasing on each sub interval [zi−1,zi] . Then

∫ ba f dx exists.

Proof: Let Īk and Īk and Ik pertain to the interval [zk−1,zk] . Then these are equal and sothere is a parition P of [a,b] including all the zk such that U ( f ,P)−L( f ,P)< ε. You justconsider an appropriate partition of [zk−1,zk] making the difference between the upper andlower sums less than ε/n for each of these sub intervals. Thus there is no space between Īand I because ε is arbitrary.

The above corollary shows that all reasonable bounded functions are integrable. Thisis really a case of the following theorem.

Theorem 7.3.10 Suppose a bounded real valued function f is integrable on [a,c]and that a < b < c. Then the restrictions of this function to [a,b] and [b,c] are integrableon these intervals and in fact, ∫ b

af dx+

∫ c

bf dx =

∫ c

af dx