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Math 332 Final Exam Fall 2013 Louiza Fouli
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Instructions:
1. Make sure you have all 11 pages (including this cover page).
2. No notes are allowed.
3. You must show su�cient work to receive credit. Correct answers without su�cient
work will receive no credit.
4. The point value of each problem occurs to the left of each problem.
5. Good luck!
Page Points Points Possible
2 8
3 9
4 9
5 10
6 8
7 16
8 10
9 10
10 10
11 10
Total 100
1
Math 332 Final Exam Fall 2013
1. (8 pts)
(a) State carefully the definition for a function to be discontinuous at a point in its
domain.
(b) State carefully the definition of a uniformly continuous function.
(c) State carefully the definition of infimum of a set.
(d) State carefully the definition of the limit of a function at a point.
2
Math 332 Final Exam Fall 2013
2. (9 pts) Let S = {a 2 R | a2 < 1}. Give answers to the following questions. You do not
need to prove your claims.
(a) Find all the lower bounds for S if they exist.
(b) Find the minimum of S if it exists.
(c) Find the infimum of S if it exists.
3
Math 332 Final Exam Fall 2013
3. (9 pts) Prove that if a, b 2 R and a < b+ ✏ for every ✏ > 0, then a b.
4
Math 332 Final Exam Fall 2013
4. (10 pts) Let ✏ > 0. For every x 2 R we define the set N✏(x) = {y 2 R | |x � y| < ✏}.The set N✏(x) is called an ✏-neighborhood of x. Let a, b 2 R such that a < b. Prove that
there exists ✏ > 0 such that N✏(a) \N✏(b) = ;.
5
Math 332 Final Exam Fall 2013
5. (8 pts) Find an interval of length 1 that contains a real root of f(x) = x
5 � x+ 1.
6
Math 332 Final Exam Fall 2013
6. (16 pts) Let f : R �! R be a function defined by f(x) =
8<
:
x
2 � 4
x� 2
when x 6= 2
4 when x = 2
(a) (8 pts) Find the limit of f at x = 1 if it exists and prove your answer.
(b) (8 pts) Is f continuous at x = 2? Prove your answer.
7
Math 332 Final Exam Fall 2013
7. (10 pts) Let An = (1 +
1n , 2) for all n 2 N. Prove that
1Si=1
An = (1, 2).
8
Math 332 Final Exam Fall 2013
8. Let f : R �! R be defined by f(x) =
⇢2 when x 2 Q1 when x 62 Q . Prove that f is discontinuous
at every x 2 R.
9
Math 332 Final Exam Fall 2013
9. (10 pts) Let S be a nonempty subset of R. Let T = {s+2 | s 2 S} be another set. Prove
that supT exists and that supT = supS + 2.
10
Math 332 Final Exam Fall 2013
10. (10 pts) Let f : (2, 7) �! R be defined by f(x) = x
2 � x + 1. Use the definition of
uniform continuity to show that f is uniformly continuous on (2, 7).
11
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