Math 554/703I - Analysis I
Test 3 – April 14, 2004
Name:
Directions: To receive credit, you must justify your statements un-
less otherwise stated. Answers should be provided in
complete sentences.
1 (20 pts)
2(10 pts)
3(10 pts)
4(10 pts)
5(15 pts)
6(10 pts)
7(15 pts)
8(10 pts)
1. Give an example of each of the following and (very) briefly justify your answer:
(a) A closed set of real numbers that is not connected.
One Soln: {0,1}is closed.
(b) A set of real numbers that is complete but not connected.
One Soln: [0,1] ∪[2,3], a subset of the real numbers is complete iff it is closed.
(c) A compact set of real numbers that is not connected.
One Soln: [0,1] ∪[2,3] is closed and bounded but is not connected.
(d) A real-valued continuous function that does not satisfy the Intermediate Value Theorem.
One Soln: Let dom(f) = {0} ∪ [1,2] and f(x) = xon that domain. The intermediate
value of 1
2is not attained.
(e) A real-valued function that is continuous at a point x0, but is not differentiable.
One Soln: The function f(x) = |x|is continuous at x0= 0, but is not differentiable there;
observe sequential limits of the difference quotient applied to the sequences {−1
n}and {1
n}.
2. Prove that each compact set is closed.
Proof: Suppose that xobe a limit point of the compact set Kwhich does not belong to K.
Consider the open sets Onwhich are the complements of the closed balls Cn={x| |x−x0| ≤ 1
n}.
Since x0/∈K, then the collection of On’s form an open cover of Kand so must have a finite
subcover. Since On⊂ On+1, then the largest set in this finite collection will cover K. Call
that set ON. But then B1
N(x0)⊂CNis disjoint from K.Contradiction, since each Bǫ(x0)
must contain an infinite number of members of K.
3. State and sketch a proof of the Heine-Borel theorem.
Soln: See course lecture notes as well as daily lecture notes.
4. a.) Define connectedness for a set of real numbers A.
Soln: See course lecture notes as well as daily lecture notes.
b.) Prove that if a continuous function fis defined on an interval I, then the range of fis an
interval.
Proof: A subset of real numbers with the standard metric is connected iff it is
an interval. Since the continuous image of a connected set is connected, then the
continuous image of an interval is an interval.
