5.7 Darboux sums and Riemann integrals 351ThusF'(x) exists when -1 =< x 5 1. As x approaches 0, F(x) oscillates between
-1 and 1 and does not have a limit, so F is not continuous at the place where
x = 0. However, F is Riemann integrable over the interval -1 5x 5 1
because F'(x) exists and is bounded over the interval and is continuous except
at one place. Thus our function F has the required properties. We could have
used the formula
(5) F(x)=sin1dt
loz tinstead of (2) to define F. It would then have been slightly easier to obtain (4)
but would not have been so easy to show that F'(0) = 0. There is a reason why
no simpler example can be given. Derivatives must have the intermediate-
value property, and no discontinuous function having the intermediate-value
property is simpler than the function 0 for which q5(0) = 0 and O(x) = sin (1/x)
when x 0 0.
7 If the unique individual that some textbooks like to call "the student"
is unable to prove that each polynomial is bounded and piecewise monotone
over each interval a 5 x 5 b, there are only three possible places to place the
blame. Is it the student? Is it the textbook? Is it the problem?
8 We have, at one time and another, seen examples of faulty applications of
the noble but frequently invalid premise that a thing T must be an element of a
set S if T is the limit of a sequence of elements of S. One old example involves
the "idea" that a circle must be a polygon because it is the limit of polygons.
Another old example involves the "idea" that a Riemann integral must be the
sum of infinitely many things because it is the limit of sums. Should we swallow
the "idea" that an irrational number must be a rational number because it is the
limit of rational numbers? dns.: No.
9 We have the possibility of extending our intellectual horizons by investing
a few minutes or a few years in study of algebras which differ from the algebra
of real numbers. The algebra of rational functions invites us to consolidate old
ideas and capture new ones. When ac, a,, ,a,,, and bo, bi, , b are
constants for which the b's are not all zero, the two polynomials P and Q for which
P(x) = ao + alx + ... + amxm, Q(x) = bo + bix + ... + bnxn
determine the rational function f for which f(x) = P(x)/Q(x) for those values
of x for which Q(x) 76 0. The sum f + g of two rational functions is the rational
function h for which h(x) = f(x) + g(x) for each x for which the sum is defined.
If c is a constant and f is a rational function, then cf is the rational function having
values cf(x). If f and g are rational functions, then fg is the rational function
having values f(x)g(x) and [unless g(x) = 0 for each x] f/g is the rational function
having values f(x)/g(x) when g(x) 0. Textbooks in modern algebra call atten-
tion to many respects in which the algebra of rational functions is like the algebra
of real numbers. Terminologies involving rings, fields, and groups facilitate dis-
cussions of these matters. Nontrivial interest in the algebra of rational functions
starts to develop when order relations are introduced in a particular special way.
We say that f < g and g > f if there is a number xo such that f(x) < g(x) and