1549901369-Elements_of_Real_Analysis__Denlinger_

7.6 The Fundamental Theorem of Calculus 405

But Ica f = -I: f and I: f = -I: f. Thus, Equation (21) becomes

• I: f =I: f - I: f , which is equivalent to

Theorem 7.6.6 (Continuity of the Integral) Suppose f is integrable on a

compact interval I , and a E I. Then the function F : I --+ JR defined by the

formula F(x) =I: f is (uniformly) continuous on I.

a x x

I

Figure 7.8

Proof. Suppose f is integrable on a compact interval I , and a E J. Then
'ix EI, f is integrable over [a, x] if a::; x (or [x, a] if x::; a) by Corollary 7.4.4,
and so the function F(x) =I: f is defined 'ix EI. Since f is integrable over I
it is bounded there, and so 3 M > 0 3 Vt EI, lf(t)I ::; M.

Then, 'ix, y EI,

IF(y) - F(x)I = II: f -I: fl

= II: fl by Theorem 7.6.5

<

{

I: If I if x ::; y

by Corollary 7.5.5

I; Iii if x > y

<

{

I: M if x ::; y

by Corollary 7.5.5

Jy rx M if x > y

= Mly - xi. (integral of a constant)