1549901369-Elements_of_Real_Analysis__Denlinger_

392 Chapter 7 a The Riemann Integral

y

y =f(x)

a b x

Figure 7.7

Proof. Exercise 14. •

*REGULATED FUNCTIONS

While each of the conditions given in (7.4.13) and (7.4.14) is sufficient to
guarantee that a function f : [a, b] ---+ JR is integrable on [a, b], it would be
helpful to have one straightforward condition for integrability that includes all
of these as special cases. We shall see that the following condition is sufficient.

Definition 7.4.15 A function f : [a, b] ---+ JR is said to be regulated on [a, b]
if 'Vxo E [a, b), lim f(x) exists, and 'Vxo E (a, b], lim f(x) exists.
X--+Xo + X--+Xo -
Given a regulated function f : [a, b] ---+ JR, and a point x 0 E [a, b], we use
the notation

f(xo-) = lim_ f(x) and f(xo+) = lim+ f(x),

X--+Xo X--+Xo

and 'Vxo E (a, b) we define the jump of f at x 0 to be

j(f,xo) = max{lf(xo+) - f(xo)I, lf(xo-) - f(xo)I, lf(xo+) - f(xo-)I}.

We define the jump of f at a and b to be

j(f,a) = lf(a+)-f(a)I and j(f,b) = lf(b-) -f(b)I.

Remark 7.4.16 Given any regulated f: [a, b] ---+JR and x 0 E [a, b], j(f, x 0 ) 2:

0. Moreover, f is continuous at x 0 {::} j(f,x 0 ) = 0.