Math 501
Introduction to Real Analysis
Instructor: Alex Roitershtein
Iowa State University
Department of Mathematics
Summer 2015
Homework #5
Due date: Monday, July 6
1. Let α and c be real numbers, c > 0, and f is defined on [−1, 1] by
α
x sin(xc ) if x 6= 0
f (x) =
0
if x = 0.
Find all the values of the parameters α and c such that:
(a) f is continuous.
(b) f 0 (0) exists.
(c) f 0 is continuous.
(d) f 00 (0) exists.
2. Let a ∈ R be a given real number. Suppose that a function f : R → R satisfies
f 0 (x) < 0 < f 00 (x)
if x < a,
f 0 (x) > 0 > f 00 (x)
if x > a.
and
Prove that f 0 (a) does not exists.
Hint: To see informally the reason why f 0 (a) doesn’t exist, draw the graph of, for example,
(x − a)2
if x < a,
f (x) =
1
1 − 1+x−a if x ≥ a.
The monotonicity of f (x) on (a, ∞) and (−∞, a) dictates f 0 (a) = 0 if the derivative exists.
But f 0 (a) = 0 means that a is a local maximum of f, and hence f 00 (0) < 0. This is impossible
since f 00 is decreasing and positive on x < a. To make a formal proof from this argument,
(a)
use the mean value theorem f (x)−f
= f 0 (yx ) and the fact that the derivative is monotone
x−a
on each side of a, to deduce that if f 0 (a) exists then
f 0 (a) = lim f 0 (yx ) = sup f 0 (x) = inf f 0 (x) = 0.
x→a
x>a
x<a
Use then the mean value theorem for the derivative
assertion f 0 (a) = 0 contradicts the assumptions.
3.
1
f 0 (x)−f 0 (a)
x−a
= f 00 (zx ) to check that the
(a) Let g(y) = arcsin y 3 . Compute g 0 (y) for y ∈ (0, 1).
(b) Suppose that f : R → R is differentiable and that there exists n ∈ N such that
f (tx) = tn f (x)
for any t > 0 and x ∈ R. Show that xf 0 (x) = nf (x) for all x ∈ R.
4. Solve Exercise 80 in Chapter 2 of the textbook.
5. Consider a function f : R → R. Suppose that
(i) f is continuous for x ≥ 0.
(ii) f 0 (x) exists for x > 0.
(iii) f (0) = 0.
(iv) f 0 is monotonically increasing.
Let
g(x) =
f (x)
,
x
x > 0.
Prove that g is monotonically increasing.
6. Let X be the set of all bounded real-valued functions on a non-empty set S.
(a) For x, y ∈ X set d(x, y) = supt∈S |x(t) − y(t)|. Show that d is a metric on X.
(b) For x ∈ X, let
f (x) = inf x(t)
t∈S
and
g(x) = sup x(t).
t∈S
Show that f and g are uniformly continuous functions from (X, d) to R.
2