Midterm Exam
Math 501
Due October 17, 2014
Instructions: This is a take-home exam. You must complete all of the
problems below and turn in your solutions in class on the due date. You may
consult only your textbook, your lecture notes, the lecture notes on the class
web page, and me (Tim).
Problem 1
A function f : (a, b) → R is called convex if for any x, y ∈ (a, b)
f (sx + ty) ≤ sf (x) + tf (y)
for all s, t ∈ [0, 1] such that s + t = 1. Prove that f is convex if and only if the
set S = {(x, y) ∈ (a, b) × R : y ≥ f (x)} is a convex subset of R2 .
Problem 2
Let α > 0. Let x1 >
√
α and define a sequence (xn ) recursively by
1
α
xn+1 =
xn +
2
xn
(1)
for n ≥ 1.
(a) Prove the inequality
√
x + y ≥ 2 xy
holds for all x, y > 0.
(b) Use√the inequality above to show that the sequence (xn ) is bounded below
by α.
(c) Prove that (xn ) is monotone decreasing. [Hint: show that xn+1 − xn ≤ 0
for all n ≥ 1.]
1
(d) Argue that (xn ) converges.
√
(e) Show that xn → α. [Hint: let L = lim xn . Take the limit of both sides as
n → ∞ in (1) and solve for L.]
Remark. The problem above gives an algorithm for approximating square roots.
Problem 3
Let S ⊂ M . Recall that the distance to S was defined as
dist(p, S) = inf{d(p, s) : s ∈ S}.
(a) Prove that for any p, q ∈ M
| dist(p, S) − dist(q, S)| ≤ d(p, q).
(b) Prove that the function f : M → R given by f (p) = dist(p, S) is uniformly
continuous.
Problem 4
Consider the set C[a, b] of continuous, real-valued functions whose domain
is the closed interval [a, b]. There are two (among others) natural metrics which
one may consider on C[a, b]. The first, the sup-metric, is one we’ve already
encountered:
d∞ (f, g) = sup{|f (x) − g(x)| : x ∈ [a, b]}.
The second, called the L1 metric, is given by
Z b
d1 (f, g) =
|f (x) − g(x)| dx.
a
Denote by Cmax the space C[a, b] equipped with the d∞ metric, and denote by
Cint the space C[a, b] equipped with the d1 metric.
In this problem, we show that the identity map I : Cmax → Cint is continuous
(and obviously a bijection) but its inverse is not continuous, which hints at the
fact that Cmax and Cint are not homeomorphic.
Let (fn ) ⊂ C[a, b] be a sequence and let f ∈ C[a, b]. We say fn → f in
Cmax if d∞ (fn , f ) → 0 as n → ∞. Likewise, fn → f in Cint if d1 (fn , f ) → 0 as
n → ∞.
(a) Let fn , f ∈ C[a, b] and suppose that fn → f in Cmax . Prove that fn → f
in Cint . Conclude that the identity map I defined above is continuous.
2
(b) Let fn (x) = (x − a)n /(b − a)n and let f (x) = 0 for all x ∈ [a, b]. Show that
fn → f in Cint but not in Cmax . Conclude that the inverse of the identity
map is not continuous.
Hint: the following easy lemma, which you may use without proof, will be
useful.
Lemma. Let f ∈ C[a, b] and let α = sup{|f (x)| : x ∈ [a, b]}. Then
Z
b
f (x) dx ≤ α(b − a).
a
Problem 5
Let f : M → N be a continuous bijection and suppose M is covering compact.
(a) Let C ⊂ M be closed. Prove directly that C is covering compact.
(b) Let C ⊂ M be closed. Prove that f (C) is closed.
(c) Prove that f −1 is continuous.
(d) Conclude that f is a homeomorphism.
Problem 6
Let f : M → R be a function and recall that its graph is the set Γ = {(x, y) ∈
M × R : y = f (x)}.
(a) Prove that if M is connected and f is continuous, then Γ is connected.
(b) Show that the converse of (a) is not true by giving an example of a function
which is not continuous but whose domain and graph are both connected.
3