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