418 Chapter 7 • The Riemann Integral
have been extended and updated by Ivan Niven in his enlightening monograph,
Irrational Numbers, [100]. In addition to proofs of our Theorems 7.6.21 and
7.6.22, Chapter 2 of Niven's book includes proofs of many additional interest-
ing results, including:
(1) For every rational number x =J 0, the trigonometric functions of x are all
irrational.
(2) Any nonzero value of an inverse trigonometric function of x is irrational
for all nonzero rational values of x.(3) For every rational number x =J 0, the hyperbolic functions of x are all
irrational.( 4) Any nonzero value of an inverse hyperbolic function of x is irrational for
all nonzero rational values of x.
(5) For all positive rational x =J l , lnx is irrational.(6) For all positive rational x =J l , logb xis irrational for any positive rational
base b =J 1 unless xm = bn for some integers m and n.In Chapter 9 of the same book, Niven also proves the deeper and more
remarkable results that these numbers are "transcendental." [Recall^13 that
a number a is transcendental if there is no polynomial p(x) with rational
coefficients such that p(a) = O.] These results may help explain the title of the
next section.
EXERCISE SET 7.6l. Prove Case 1 of the proof of Theorem 7 .6.5.
Select two of the relative positions of a, b, and c not treated in Theorems
7.4.2 or 7.6.5 and prove that theorem for those two cases.
Show that the signum function sgn(x) = { l~I if x =JO} is integrable
0 if x = 0
on [-1, 1], yet has no antiderivative on [-1, l]. [Thus, a function can be
integrable on an interval without having an antiderivative there. For a
more complicated example, see Exercise 4.]
In Exercise 7.4.18 we showed that Thomae's function Tis integrable on
[0,1], even though it is discontinuous at every rational number. Prove that
T has no antiderivative on any subinterval of [O, l]. [Hint: Use Theorem
6.3.7.]
See Exercise 2.8.17.