846 APPENDIX A. THE THEORY OF THE RIEMANNN INTEGRAL∗

Theorem A.2.8 (Riemannn criterion) f ∈R (Rn) if and only if for all ε > 0 there exists agrid G such that

UG ( f )−LG ( f )< ε.

Proof: If f ∈R (Rn), then I ( f ) = I ( f ) and so there exist grids G and F such that

UG ( f )−LF ( f )≤ I ( f )+ε

2−(

I ( f )− ε

2

)= ε.

Then letting H = G ∨F , Lemma A.2.4 implies

UH ( f )−LH ( f )≤UG ( f )−LF ( f )< ε.

Conversely, if for all ε > 0 there exists G such that

UG ( f )−LG ( f )< ε,

thenI ( f )− I ( f )≤UG ( f )−LG ( f )< ε.

Since ε > 0 is arbitrary, this proves the theorem. ■

A.3 Basic PropertiesIt is important to know that certain combinations of Riemannn integrable functions areRiemannn integrable. The following theorem will include all the important cases.

Theorem A.3.1 Let f ,g ∈R (Rn) and let φ : K→ R be continuous where K is a compactset in R2 containing f (Rn)×g(Rn). Also suppose that φ (0,0) = 0. Then defining

h(x)≡ φ ( f (x) ,g(x)) ,

it follows that h is also in R (Rn).

Proof: Let ε > 0 and let δ 1 > 0 be such that if (yi,zi) , i = 1,2 are points in K, such that|z1− z2| ≤ δ 1 and |y1− y2| ≤ δ 1, then

|φ (y1,z1)−φ (y2,z2)|< ε.

Let 0 < δ < min(δ 1,ε,1). Let G be a grid with the property that for Q ∈ G , the diameterof Q is less than δ and also for k = f ,g,

UG (k)−LG (k)< δ2. (1.5)

Then defining for k = f ,g,

Pk ≡ {Q ∈ G : MQ (k)−mQ (k)> δ} ,

it followsδ

2 > ∑Q∈G

(MQ (k)−mQ (k))v(Q)≥