Math 501
Introduction to Real Analysis
Instructor: Alex Roitershtein
Iowa State University
Department of Mathematics
Summer 2015
Homework #6
Due date: Friday, July 10
1. Suppose that f : R → R is continuous on [a, b] and P is a partition of [a, b]. Show that
Rb
there exists a Riemann sum of f over P that equals a f (x)dx.
2. Compute supP L(P, fi ) and inf P U (P, fi ) on [0, 1] for i = 1, 2, and the following two
functions defined on [0, 1] :
x
if x ∈ Q
1 if x ∈ Q
f1 (x) =
and
f2 (x) =
−x if x 6∈ Q
x if x 6∈ Q.
3.
(a) Prove that if f : R → R is absolutely integrable on [0, +∞) then
Z ∞
lim
f (xn )dx = 0.
n→∞
1
(b) Suppose that f : R → R is continuously differentiable and one-to-one on [a, b]. Prove
that
Z f (b)
Z b
f −1 (x)dx = bf (b) − af (a).
f (x)dx +
a
f (a)
where f −1 is the inverse function of f.
4. Suppose that f : R → R is bounded and f 2 is integrable on [a, b]. Does it follow that f
is integrable on [a, b]?
5. Suppose that f : R → R is continuous and non-negative on [0, 1]. Show that if
then f (x) = 0 for every x ∈ [0, 1].
6. Solve Exercise 37 in Chapter 2 of the textbook.
1
R1
0
f (x) = 0,