182 CHAPTER 7. THE INTEGRAL

Now consider the claim about an integrable function being changed at finitely manypoints. Let f be the integrable function. Let the finitely many points be z1,z2, ...,zr listedin order. Also let f̂ be the modified function and let

M ≡ sup(max

(∣∣ f̂ (x)∣∣ , | f (x)|) ,x ∈ [a,b]),

m ≡ inf(min

(−∣∣ f̂ (x)∣∣ ,−| f (x)|

),x ∈ [a,b]

)Let P ≡ {x0,x1, ...,xn} be a partition which contains all the finitely many points and

suppose also that |xk − xk−1|< δ where 2(M−n)rδ < ε

2 and also

U ( f ,P)−L( f ,P)<ε

2Let the zi be xki . Then

U(

f̂ ,P)−L

(f̂ ,P)≤U ( f ,P)−L( f ,P)

+(M−m)r

∑i=1

(xki+1 − xki

)+(M−m)

r

∑i=1

(xki − xki−1

)<

ε

2+2(M−n)rδ < ε

Since ε is arbitrary, this shows there is no gap between the upper and lower integrals andso f̂ is also integrable.

Corollary 7.3.14 Every piecewise continuous function is integrable and if f is piece-wise continuous and f = fk on (zk−1,zk) where fk is continuous on [zi−1,zi] , then∫ b

af (x)dx =

r

∑k=1

∫ zk

zk−1

fk (x)dx

Proof: From what was just shown, f is integrable on each [zk−1,zk] because it equalsa continuous function except maybe at the end points. Also, from induction and Theorem7.3.12 f is integrable on [a,b] and

∫ ba f (x)dx = ∑

rk=1

∫ zkzk−1

f (x)dx = ∑rk=1

∫ zkzk−1

fk (x)dx.Now here is a one version of the fundamental theorem of calculus.

Theorem 7.3.15 Let f (x) = F ′ (x) for x ∈ (a,b) and F is continuous on [a,b]. Alsosuppose f is integrable. Then

∫ ba f (x)dx = F (b)−F (a).

Proof: There is a partition P such that U ( f ,P)−L( f ,P) < ε . Then letting xk denotethe points of P in the usual way, by the mean value theorem, there exists zk ∈ (xk−1,xk)such that

F (b)−F (a) =n

∑k=1

F (xk)−F (xk−1)

=n

∑k=1

f (zk)(xk − xk−1) ∈ [L( f ,P) ,U ( f ,P)]

an interval of length no more than ε . But also∫ b

a f (x)dx is in this same interval and so∣∣∣∣(F (b)−F (a))−∫ b

af (x)dx

∣∣∣∣< ε

Since ε is arbitrary, this shows∫ b

a f (x)dx = F (b)−F (a).At this point this integral is seen to be a generalization of the earlier integral defined

according to the fundamental theorem of calculus. Next consider functions of RiemannDarboux integrable functions.