ANALYSIS I: PROBLEM SET # 6
DUE FRIDAY, 15 APRIL
Ra
Exercise 42. Consider a connected open subset U ⊂ R containing two points x and y. Suppose f : U
differentiable function (but not necessarily continuously differentiable). Show that the function f 0 : U
R takes
every value between f 0 (x) and f 0 (y). Deduce that the function f : R
R given by

if x > 0;
1
f (x) := 0
if x = 0;

−1 if x < 0;
cannot be the derivative of any real-valued function on R.
R that is not monotonic in any neighborhood
Exercise 43. Find an example of a differentiable function f : R
of 0 but whose derivative at 0 is nevertheless positive.1
Definition. Suppose U ⊂ R an open set, and suppose k a natural number. Recall that a function f : U
R is said
to be of class C k if for every natural number n ≤ k, the n-fold derivative exists and is continuous on U . We denote
by C k (U ) the set of such functions.2
A function f : U
R is said to be smooth or infinitely differentiable or of class C ∞ if it is of class C k for every
natural number k. We denote by C ∞ (U ) the set of such functions.
Exercise 44. Suppose n a positive integer. Consider the function fn : R
R defined by
n
fn (x) := |x| .
What is the largest natural number k such that fn ∈ C k (R)?
Definition. Suppose X and Y metric spaces, and suppose α a nonnegative real number; then a map f : X
Y is
said to be Hölder of exponent α or, more briefly, α-Hölder if there is a nonnegative real number M such that for any
x, y ∈ X ,
d ( f (x), f (y)) ≤ M d (x, y)α .
We say that f Lipschitz if it is Hölder of exponent 1.
Exercise 45. Suppose X and Y metric spaces, and suppose f : X
Y a map. Note that f is 0-Hölder if and only
if f is bounded, and show that if f is α-Hölder for some α > 0, then f is uniformly continuous. Is every uniformly
continuous map between two metric spaces Hölder of some exponent α ≥ 0?
Exercise 46. Suppose E ⊂ R a connected set. Show that a map f : E
if f is constant.
R is Hölder of exponent α > 1 if and only
Notation. Suppose X a metric space. For any α ∈ [0, 1], denote by C 0,α (X ) the set of α-Hölder functions X
Exercise 47. Suppose Ω ⊂ R a bounded set. Show that if 0 ≤ α < β ≤ 1, then C
to show that this containment is strict.
0,β
(Ω) ⊂ C
0,α
(Ω); give an example
Exercise? 48. Suppose Ω ⊂ R a bounded set, and suppose 0 ≤ α ≤ 1. For any f ∈ C 0,α (Ω), write
|| f ||0,α := sup | f (x)| +
x∈Ω
sup
x,y∈Ω, x6=y
| f (x) − f (y)|
|x − y|α
,
1Hint: find a function f with the property that f 0 takes both positive and negative values in every neighborhood of 0.
2More generally, this definition makes sense if U is an open set of R m .
1
R.
2
DUE FRIDAY, 15 APRIL
a nonnegative real number. Show that the formula
d0,α ( f , g ) := ||g − f ||0,α
defines a metric on C 0,α (Ω). Is the metric space (C 0,α (Ω), d0,α ) complete?
Exercise 49. Suppose U ⊂ R a connected open set, and suppose that f : U
R is a continuous function that is
0
0,1
R is bounded. Deduce that if
differentiable at every point of U . Show that f ∈ C (U ) if and only if f : U
f ∈ C 1 (U ), then its restriction to a bounded subset Ω ⊂ U is Lipschitz.3
Exercise 50. We have concentrated on situations in which the derivative of a function f : R
we may say that f has infinite derivative4 at a point x ∈ R if the function
R is finite. However,
| f (x + h) − f (x)|
|h|
increases without bound as h → 0. Give an example of a continuous function with an infinite derivative at 0. Is it
true that if f has infinite derivative at a point, then it is continuous there?
Exercise? 51. Read the statements and proofs of Theorem 7.17, Theorem 8.1, and the Corollary to Theorem 8.1 in
Rudin. Note that these results imply that exp ∈ C ∞ (R).
(a) Show that the function φ: R
R given by
(
€
Š
exp − x12
if x > 0;
φ(x) :=
0
if x ≤ 0.
is smooth. This smooth function has the property that f (x) vanishes for negative x, is positive and monotonic
for positive x, and converges to 1 as x → ∞.
R given by
(b) Similarly, show that the function ψ: R
(
€
Š
1
exp − 1−x
if |x| < 1;
2
ψ(x) :=
0
if |x| ≥ 1.
is smooth. This smooth function has the property that it vanishes outside the interval (−1, 1) and is positive
inside (−1, 1).
(c) Using the ideas from the previous two parts, construct, for any closed interval I and any open interval U
containing I , a function ρI ,U ∈ C ∞ (R) with all the following properties.5
(1) For any x ∈ R, one has 0 ≤ ρI ,U (x) ≤ 1.
(2) For any x ∈ R − U , one has ρI ,U (x) = 0.
(3) For any x ∈ I , one has ρI ,U (x) = 1.
(d) Conclude that for any function f ∈ C k (R), any closed interval I and any open interval U containing I , there is
a function fI ,U ∈ C k (R) such that
(1) For any x ∈ R, one has | fI ,U (x)| ≤ | f (x)|.
(2) For any x ∈ R − U , one has fI ,U (x) = 0.
(3) For any x ∈ I , one has fI ,U (x) = f (x).
3If you like, you may prove the following more general version instead. Suppose Ω ⊂ R m a convex open subset, and suppose that f : Ω
is a continuous function that is differentiable at every point of Ω. Show that f is Lipschitz if and only if
sup
x∈Ω, ξ ∈S m−1
Rn
||d x f (ξ )|| < ∞,
where S m−1 is the sphere {ξ ∈ R m | ||ξ || = 1}. Deduce that if f is continuously differentiable, then its restriction to a compact subset K ⊂ Ω is
Lipschitz. In fact, all this remains true when R m and Rn are replaced by Banach spaces.
4Please do not confuse having infinite derivative at 0 with being infinitely differentiable at 0; these are very different!
5For an additional challenge, construct a function ρ
∞
m
m
I ,U ∈ C (R ) with these properties, where I is a compact subspace of R and U is an
open set containing I .