1549901369-Elements_of_Real_Analysis__Denlinger_

390 Chapter 7 • The Riemann Integral

The next theorem is really rather remarkable, and perhaps unexpected.

The result shows the power of the way of thinking we have developed in this

course.

Theorem 7.4.9 Changing the values of a function at finitely many

points of [a, b] affects neither its integrability nor the value of its integral there.

More precisely, suppose f ,g: [a,b] __,~are bounded on [a,b] and f(x) = g(x)

for all but finitely many points of [a, b]. Then f is integrable on [a, b] if and only

if g is integrable on [a, b].^9 Moreover, in case of integrability, J: f = J: g.

Proof. Suppose f, g: [a, b] __,~are bounded on [a, b] and f(x) = g(x) for

all x E [a,b] except at the points {x 1 , x2, · · · , xn}, where x 1 < X2 < · · · < Xn.

Let xo =a and Xn+l = b. Then P = {xo, X1, x2, · · · , Xn, Xn+d is a partition of

[a, b], and on each subinterval [xi-l, xi] created by this partition, f and g agree

except on the endpoints. Apply Corollary 7.4.8 to each [xi-i, xi]. Corollaries

7.4.3 and 7.4.6 yield the desired conclusion. •

Thus, a function can have discontinuities at, for example, a million points

of [a, b] and still be integrable there. It is natural to ask whether it is possible

for a function to have discontinuities at an infinite number of points of [a, b] and

still be integrable there. Remembering Dirichlet's function, Example 7.2.10, one

might expect t he answer to be no. However, the following remarkable example

shows that the answer is yes.

Example 7.4.10 (A function with infinitely many points of discontinuity in

an interval on which it is integrable.) Define f: [O, 1] __,~by

f ( x) = { 1 if x = 1 / n for some n E N, }.

0 otherwise

Then, for each 0 < c < d < 1, f equals the constant function g(x) = 0,

except for at most finitely many points. So, by Theorem 7.4.9, f is integrable

on [c, d] and t f = t g = 0. Therefore, by Theorem 7.4.7, f is integrable on

[a, b] and f 01 f = lim J,^1 - h f = 0. However, f is discontinuous on the infinite

h-->O+ i

set { ~ : n E N}. D

For examples of functions that are integrable on [a, b] yet discontinuous on

a dense subset of [a, b], see Exercises 18 and 1 9.

9. In fact, we s ha ll say that f is integrable on [a , b] even if it is not defined at some points
   where it differs from an integrable function g. For example, we shall say tha t JxJ/ x is integrable
   on [-1, 1] even though it is not defined at 0.