Kenneth Kuttler

314 CHAPTER 13. THE GENERALIZED RIEMANN INTEGRAL

As ηn→ 0, that last term f (b)(

F (b−)−F(

xnmn−1

))+ f (a)(F (xn

1)−F (a+)) convergesto 0 and so the end result is

(F (a+)−F (a)) f (a)+∫ b

af dF̂ + f (b)(F (b)−F (b−))

Thus this must be the generalized Riemann integral.

Procedure 13.3.5 Suppose

1. Suppose F is increasing and may have jumps at a and b but no jumps on (a,b).

2. Let F̂ (a)≡ F (a+) , F̂ (b)≡ F (b−)

3. Let f ∈ R([a,b] , F̂

)Then f ∈ R∗ [a,b] and∫ b

af dF = (F (a+)−F (a)) f (a)+

∫ b

af dF̂ + f (b)(F (b)−F (b−))

This says you add the weighted jumps at the end points to the ordinary Riemann inte-gral taken with respect to F̂ which is continuous on [a,b]. What if you had two intervals[a,b] , [b,c] and similar conditions holding on each. Then you would obtain∫ c

af dF = (F (a+)−F (a)) f (a)+

∫ b

af dF̂ + f (b)(F (b+)−F (b−))

+∫ c

bf dF̂ + f (c)(F (c)−F (c−))

The details are left to you. Note how you pick up the whole jump at b. You could of coursestring together as many of these as you want. In each of the two integrals, F̂ is adjusted tobe continuous on the corresponding closed interval. What is new here? The thing which isnew is that f does not need to be continuous at points where F has a jump. It is not evenclear that the ordinary Rieman Stieltjes integral exists.

13.4 Integrals of DerivativesConsider the case where F (t) = t. Here I will write dt for dF. The generalized Riemannintegral does something very significant which is far superior to what can be achieved withother integrals. It can always integrate derivatives. Suppose f is defined on an interval,[a,b] and that f ′ (x) exists for all x ∈ [a,b], taking the derivative from the right or left at theendpoints. What about the formula∫ b

af ′ (t)dt = f (b)− f (a)? (13.15)

Can one take the integral of f ′? If f ′ is continuous there is no problem of course. However,sometimes the derivative may exist and yet not be continuous. Here is a simple example.

Example 13.4.1 Let

f (x) =

{x2 sin

(1x2

)if x ∈ (0,1]

0 if x = 0.

You can verify that f has a derivative on [0,1] but that this derivative is not continuous.

314 CHAPTER 13. THE GENERALIZED RIEMANN INTEGRALAs 1), — 0, that last term f (b) (F (b—)—F (x )) + f (a) (F (x) — F (a+)) convergesmn—1to 0 and so the end result is(F (at) F(a) Fla)+ [sah +£00)(F )—F D-)Thus this must be the generalized Riemann integral.Procedure 13.3.5 suppose1. Suppose F is increasing and may have jumps at a and b but no jumps on (a,b).2. Let F (a) = F (a+) ,F (b) = F (b-)3. Let f € R((a,b],F)Then f € R* [a,b] and[far = (en —F(@)s(a)+ [sah +10)F FO)This says you add the weighted jumps at the end points to the ordinary Riemann inte-gral taken with respect to F which is continuous on [a,b]. What if you had two intervals(a, b] , [b,c] and similar conditions holding on each. Then you would obtain[tar = (Fen -F@y sat [ Fah + $0) (F (OH)—F 0)+ [ fa +F(0) (F (c)—F(c-))The details are left to you. Note how you pick up the whole jump at b. You could of coursestring together as many of these as you want. In each of the two integrals, F is adjusted tobe continuous on the corresponding closed interval. What is new here? The thing which isnew is that f does not need to be continuous at points where F has a jump. It is not evenclear that the ordinary Rieman Stieltjes integral exists.13.4 Integrals of DerivativesConsider the case where F (t) = t. Here I will write dt for dF. The generalized Riemannintegral does something very significant which is far superior to what can be achieved withother integrals. It can always integrate derivatives. Suppose f is defined on an interval,[a,b] and that f’ (x) exists for all x € [a,b], taking the derivative from the right or left at theendpoints. What about the formulab[ f@a=F0)-Fa? (13.15)aCan one take the integral of {’? If f’ is continuous there is no problem of course. However,sometimes the derivative may exist and yet not be continuous. Here is a simple example.Example 13.4.1 Let_ J x’sin o if x € (0, 1]roy={ oe)You can verify that f has a derivative on [0,1] but that this derivative is not continuous.