Calculus: Analytic Geometry and Calculus, with Vectors

4.2 Riemann sums and integrals 215

does not exist, we look briefly at an example. Let f be the dizzy dancer
function D, defined over the interval 0= x< 1, for which

(4.252)

J D(x) = 0 (x irrational)

D(x) = 1 (x rational).

It is clear that, whatever the partition P of the interval 0 < x S 1 may
be, the Riemann sum

(4.253) D(tk) Otk

k=1

has the value 0 if the numbers ti , t2 , ,t are all irrational and has
the value 1 if the numbers ti, t27 , t are all rational. It follows
from this that there is no number I such that to each positive number e
there corresponds a positive number S such that (4.23) holds whenever

1PI < S. This shows that the symbol fo1 D(t) dt has no meaning or that

fo

1
D(t) dt does not exist.
If we suppose, as above, that f is a function which is bounded over an
interval, then the following theorem shows that the answer to the ques-
tion whether f is integrable over the interval depends only upon the set
of discontinuities of f.
Theorem 4.26 4 function f is Riemann integrable over an interval if
and only if it is bounded and the set of discontinuities off which lie in the
interval has Lebesgue measure zero.
This theorem is proved in modern textbooks that fully earn the right
to be called textbooks on advanced calculus. The proof is, from our
present point of view, both long and difficult, and we do not need to
know anything about it. Moreover, we do not need to understand the
theorem, but we should not be injured by taking a hasty look at Figure
4.261 and making a modest attempt to understand one of the definitions

a (^11141315) I I6 b
Figure 4.261
which has fundamental importance in more advanced mathematics.
A set D of points on a line is said to have Lebesgue measure 0 if to each
e > 0 there corresponds a collection 11, I2, 13, of intervals such that
each point of D lies in at least one of these intervals and, for each n = 1,
2, 3, ., the sum of the lengths of the first n intervalsis less than e.
Sometimes it is very easy to show that a given set D has Lebesgue measure
0 by showing that if e > 0, then there exist intervals I1, 12, 13. such
that each point of D is in at least one of these intervals and, moreover,
the length of I1 is less than e/2, the length of I2 is less than e/22, the length
of I3 is less than e/23, etcetera. The collection of intervals may be a