9.3. FUNDAMENTAL DEFINITIONS AND PROPERTIES 201

This is possible because either f or g is continuous at ci. Note there are at most r of theseintervals. The validity of the above inequality only depends on

∥∥P̃∥∥ so to eliminate possible

cases, assume that none of the yi equal any of the finitely many points in the list exceptpossibly yn and y0 if there is a discontinuity at an end point. Then there are m≤ r+1 closedintervals

{I j}m

j=1 which remain, other than these special ones which contain an exceptionalpoint and on each of these intervals, f is continuous. Hence denoting as U ( f ,Pj) an uppersum corresponding to a partition Pj of I j and L( f ,Pj) defined similarly, we can choose Pjon I j such that U ( f ,Pi)−L( f ,Pj)<

ε

5(r+1) . Letting P be a partition of [a,b] consisting ofthe yk along with the points of each Pj, it follows that

U ( f ,P)−L( f ,P)<m

∑j=1

(U ( f ,Pi)−L( f ,Pj))+ r (Mk−mk)(g(yk)−g(yk−1))

≤ ε

5(r+1)m+

ε

5<

2ε

5< ε

Proposition 9.3.16 Suppose f ∈R([a,b] ,g) where g is an increasing function. If f = f̂except at finitely many points {z1, · · · ,zn} at which g is continuous, then f̂ ∈R([a,b] ,g) and∫

f̂ dg =∫

f dg.

Proof: By assumption, there exists P such that U ( f ,P)−L( f ,P)< ε

4 . Then by addingin more points to P by including points on either side of the exceptional points and using thecontinuity of g at these exceptional points, we can pick these extra points close enough tothe exceptional points such that if P̂ consists of the new partition with the new points addedin,∣∣U ( f̂ , P̂

)−U ( f ,P)

∣∣< ε

4 and∣∣L( f̂ , P̂

)−L( f ,P)

∣∣< ε

4 . Therefore, U(

f̂ , P̂)−L(

f̂ , P̂)<

3ε

4 showing that f̂ is also in R([a,b] ,g) . Also, the two integrals are between all upper andlower sums. Thus these integrals are equal because if

∫f̂ dg≥

∫f dg,∣∣∣∣∫ f̂ dg−

∫f dg∣∣∣∣≤U

(f̂ , P̂)−L( f ,P)≤U

(f̂ , P̂)−L

(f̂ ,P)+

ε

4< ε

Since ε is arbitrary, this shows the two are equal. It works the same if∫

f dg≥∫

f̂ dg.In case g(x) = x so you are considering the Riemann integral, this theorem is more gen-

eral than the one which says that piecewise continuous functions are Riemann integrable.It is more general because you could have a function which is continuous except at finitelymany points but maybe the limit of the function from one side or another does not evenexist. A piecewise continuous function is defined next.

Definition 9.3.17 A bounded function f : [a,b]→R is called piecewise continuousif there are points zi such that a = z0 < z1 < · · · < zn = b and continuous functions gi :[zi−1,zi]→ R such that for t ∈ (zi−1,zi) ,gi (t) = f (t).

I think that the case of piecewise continuous functions is certainly the case of mostinterest, however. What happens is that the right and left limits of f exist at each of theexceptional points. Thus we can speak of

limx→zi+

f (x)≡ f (zi+) , and limx→zi−

f (x)≡ f (zi−)

where the first limit is taken from the right and the second limit from the left. On [zi−1,zi] ,the function gi equals f except at the endpoints it is the right or left limit of f .