4.4. PROPERTIES OF THE INTEGRAL 53

=U ( f ,P1)−L( f ,P1)+U ( f ,P2)−L( f ,P2)< ε/2+ ε/2 = ε. (4.4.9)

Thus, f ∈ R([a,c]) by the Riemann criterion and also for this partition,

∫ b

af (x) dF +

∫ c

bf (x) dF ∈ [L( f ,P1)+L( f ,P2) ,U ( f ,P1)+U ( f ,P2)]

= [L( f ,P) ,U ( f ,P)]

and ∫ c

af (x) dF ∈ [L( f ,P) ,U ( f ,P)] .

Hence by 4.4.9,∣∣∣∣∫ c

af (x) dF−

(∫ b

af (x) dF +

∫ c

bf (x) dF

)∣∣∣∣<U ( f ,P)−L( f ,P)< ε

which shows that since ε is arbitrary, 4.4.8 holds. This proves the theorem.

Corollary 4.4.5 Let F be continuous and let [a,b] be a closed and bounded interval andsuppose that

a = y1 < y2 · · ·< yl = b

and that f is a bounded function defined on [a,b] which has the property that f is eitherincreasing on

[y j,y j+1

]or decreasing on

[y j,y j+1

]for j = 1, · · · , l−1. Then f ∈ R([a,b]) .

Proof: This follows from Theorem 4.4.4 and Theorem 4.3.2.The symbol,

∫ ba f (x) dF when a > b has not yet been defined.

Definition 4.4.6 Let [a,b] be an interval and let f ∈ R([a,b]) . Then

∫ a

bf (x) dF ≡−

∫ b

af (x) dF.

Note that with this definition,∫ a

af (x) dF =−

∫ a

af (x) dF

and so ∫ a

af (x) dF = 0.

Theorem 4.4.7 Assuming all the integrals make sense,

∫ b

af (x) dF +

∫ c

bf (x) dF =

∫ c

af (x) dF.