1549901369-Elements_of_Real_Analysis__Denlinger_

408 Chapter 7 • The Riemann Integral

F(x) - F(xo). ,

Therefore, lim = f(xo). That is, F (xo) = f(xo). •

x-+xo X - X o

Example 7.6.9 Consider the function f : [O, 10] -> ~' where f = X[2,5] is

{

Oifx<2 }

the function defined in Example 7.6.7, f(x ) = 1 if 2::;: x::;: 5. We showed

Oifx> 5

{

o ifx<2 }

therethatF(x)=f 0 xf= x - 2if2:S:x:S:5 ·

3 if x > 5

{

Oifx<2 }

Note that F'(x) = 1 if 2 < x < 5. Thus, Fis differentiable everywhere

0 if x > 5

in [O, 10] except at 2 and 5, which are the points of [O, 10] where f is not

continuous. And, at every x where f is continuous, F'(x) = f(x). D

Remark 7.6.10 The Fundamental Theorem of Calculus Second Form says

that in any interval I on which f is integrable,

at every point x in I where f is continuous. In other words, differentiation

undoes integration of continuous functions: differentiation is a kind of inverse

of integration, for continuous functions.

Remark 7.6.11 Hereafter, we shall use the following abbreviations:

FTC-I will denote the Fundamental Theorem of Calculus, First Form, and

FTC-II will denote the Fundamental Theorem of Calculus, Second Form.

INDEFINITE INTEGRALS, DIFFERENTIALS,

AND SUBSTITUTION

Definition 7.6. 12 The symbol J f(x)dx is used to represent an antiderivative

off over some domain. That is, it is a function F(x) such that F'(x) = f(x)

for all x in that domain. J f(x)dx is called the indefinite integral off for

the given domain.