1549901369-Elements_of_Real_Analysis__Denlinger_

406 Chapter 7 • The Riemann Integral

E:

Let E: > 0. Choose f> = M. Then f> > 0, and Vx, y E J ,

IY - xi < f> ==? IF(y) - F(x)I ::::; Mly - xi < M · o = E:.

Therefore, Fis uniformly continuous on I. •

Example 7.6.7 Consider the function f = X[ 2 , 5 1, the characteristic function

{

0 if x < 2 }
of [2, 5], defined by f(x) = 1 if 2:::;: x:::;: 5. Then

y

2

0 if x > 5

x { 0 if x < 2 }

F(x) = 1 f = x - 2 if 2:::;: x:::;: 5.

y = j(x)

' i -~, R(xl 1 I ·,':

(a) y = f(x)

3 if x > 5

x

Figure 7.9

y

3

2

y=F(x)

2

(b) y = F(x)

x

The purpose of Example 7.6.7 is to illustrate that Fis continuous even at
points where f is not. Notice that even though f is discontinuous at 2 and 5,
F is continuous on the entire interval ( -oo, +oo). (See Figure 7.9.)

We are now ready for the second form of the Fundamental Theorem of
Calculus, which sheds more light on the connection between integration and
differentiation. Specifically, it shows that under certain circumstances, these
two processes are inverses of each other.

Theorem 7.6.8 (Fundamental Theorem of Calculus, Second Form)
Suppose f is integrable on a compact interval I , and a E I. Define the function
F on I by the formula F(x) = J: f. (See Figure 7.8.) Then F is differen-
tiable at every point x 0 E J^0 at which f is continuous; moreover, at any such
xo, F'(xo) = f(xo).