13.4. INTEGRALS OF DERIVATIVES 315

The fact that derivatives are generalized Riemann integrable depends on the followingsimple lemma called the straddle lemma by McLeod [22].

Lemma 13.4.2 Suppose f : [a,b]→ R is differentiable. Then there exist δ (x)> 0 suchthat if u≤ x≤ v and u,v ∈ (x−δ (x) ,x+δ (x)), then∣∣ f (v)− f (u)− f ′ (x)(v−u)

∣∣< ε |v−u| .

Proof: Consider the following picture.u x v

From the definition of the derivative, there exists δ (x)> 0 such that if |v− x|, |x−u|<δ (x), then ∣∣ f (u)− f (x)− f ′ (x)(u− x)

∣∣< ε

2|u− x|

and ∣∣ f ′ (x)(v− x)− f (v)+ f (x)∣∣< ε

2|v− x|

Now add these and use the triangle inequality along with the above picture to write∣∣ f ′ (x)(v−u)− ( f (v)− f (u))∣∣< ε |v−u| .

The next proposition says 13.15 makes sense for the generalized Riemann integral.

Proposition 13.4.3 Suppose f : [a,b]→ R is differentiable. Then f ′ ∈ R∗ [a,b] and

f (b)− f (a) =∫ b

af ′dx

where the integrator function is F (x) = x.

Proof: Let ε > 0 be given and let δ (x) be such that the conclusion of the above lemmaholds for ε replaced with ε/(b−a). Then let P = {(Ii, ti)}n

i=1 be δ fine. Then using thetriangle inequality and the result of the above lemma with ∆xi = xi− xi−1,∣∣∣∣∣ f (b)− f (a)−

n

∑i=1

f ′ (ti)∆xi

∣∣∣∣∣ =

∣∣∣∣∣ n

∑i=1

f (xi)− f (xi−1)− f ′ (ti)∆xi

∣∣∣∣∣≤

n

∑i=1

ε/(b−a)∆xi = ε.

With this proposition there is a very simple statement of the integration by parts for-mula, the product rule gives a very simple version of integration by parts.

Corollary 13.4.4 Suppose f ,g are differentiable on [a,b]. Then f ′g ∈ R∗ [a,b] if andonly if g′ f ∈ R∗ [a,b] and in this case,

f g|ba−∫ b

af g′dx =

∫ b

af ′gdx