Math 414: Analysis I
Homework 10
Due: April 18, 2014
Name:
The following problems are for additional practice and are not to be turned in: (All
problems come from Basic Analysis, Lebl )
Exercises: 3.4.3, 3.4.4, 3.4.7, 4.1.2, 4.1.5, 4.1.6, 4.1.9, 4.1.11
Turn in the following problems.
1. Exercise 3.4.8 in Basic Analysis, Lebl
2. Exercise 4.1.1 in Basic Analysis, Lebl
3. Show that the function f (x) =
1
for x ∈ R is uniformly continuous on R.
1 + x2
4. (a) Show that if f and g are uniformly continuous on S ⊂ R and if they are both
bounded on S, then their product f g uniformly continuous on R.
(b) It turns out that the condition that both f and g are bounded on S is necessary
for the product f g to be uniformly continuous. To see why, let f (x) = x and
g(x) = sin(x). Show that both f and g are uniformly continuous on R, but
that their product is not uniformly continuous on R.
5. Show that if f is continuous on [0, ∞) and uniformly continuous on [a, ∞) for some
positive constant a, then f is uniformly continuous on [0, ∞).
6. (a) Suppose that f : S → R is a uniformly continuous function on S ⊂ R. Prove
that if (xn ) is a Cauchy sequence, then f (xn ) is a Cauchy sequence in R. Hint:
Be careful not to use the Cauchy convergence criterion here. While it is true
that if (xn ) is Cauchy then it converges, the sequence could converge to a point
c outside of S. Thus f (c) may not be defined. You should be able to prove this
using only the definition of Cauchy sequences and uniform continuity.
1
(b) Using the result in (a), prove that f (x) = is not uniformly continuous on
x
S = (0, 1).
7. Not every uniformly
continuous function is a Lipschitz continuous function. Prove
√
that f (x) = x is uniformly continuous on S = [0, ∞), but is not Lipschitz continuous on S.
8. If r > 0 is a rational number, let f : R → R be defined by f (x) = xr sin(1/x) for
x 6= 0, and f (0) = 0. Determine those values of r for which f 0 (0) exist.
9. If f : S → R, where S is an interval and c ∈ S, prove that
f (x) − f (c)
x→c
x−c
lim
exists if and only if
1
Math 414: Analysis I
Homework 10
Due: April 18, 2014
f (x + h) − f (x)
h→0
h
lim
exists. If so, the limits are equal.
2