MA362 Final Exam Study Guide

The final examination will be held Thursday, May 8th at 9:00 AM in Duffy 205.

The exam will consist of three sections:

A matching section with terms definitions
Two proofs you have seen before. Two of the following four proofs will appear on the exam:
- Prove the Interior Extremum Theorem (Theorem 5.2.6)
- Prove that if a power series converges at z, it converges for any x with |x|<|z| (Theorem 6.5.1)
- Prove that a bounded function f is integrable on [a,b] if and only if for every epsilon>0, there exists a partiton P such that U(f,P)-L(f,P) (Theorem 7.2.8)
- Prove that if a function f is continuous on [a,b] then it is integrable. (Theorem 7.2.9)
Two proofs you have not seen before, chosen from a list of four possibilities that will be supplied.

The terms will be taken from the following list:

Term Text Reference

connected set Definition 3.4.4

disconnected set

separated sets

totally disconnected set Exercise 3.4.8

perfect set Definition 3.4.1

isolated point Definition 3.2.6

limit point Definition 3.2.4

dense subset P.95: G is dense in R if every element of R is a limit point of G

nowhere dense set Definition 3.5.3

closure of a set Definition 3.2.11

open set Definition 3.2.1

closed set Definition 3.2.7

compact set Definition 3.3.1

bounded set Definition 3.3.3

G-delta set Definition 3.5.1

F-sigma set

open cover Definition 3.3.6

finite subcover

function limit Definition 4.2.1

continuous function Definition 4.3.1

uniformly continuous function Definition 4.4.5

right hand limit Definition 4.6.2

left hand limit

removeable discontinuity Exercise 4.6.3

jump discontinuity

essential discontinuity

monotone function Definition 4.6.1

increasing function

decreasing function

alpha-continuous function Definition 4.6.5

pointwise convergence Definition 6.2.1B

uniform convergence Definition 6.2.3

partition Definition 7.2.1

upper sum

lower sum

refinement Definition 7.2.2

upper integral Definition 7.2.5

lower integral

Riemann-integrable Definition 7.2.7

measure zero Definition 7.6.1

Term	Text Reference
connected set	Definition 3.4.4
disconnected set
separated sets
totally disconnected set	Exercise 3.4.8
perfect set	Definition 3.4.1
isolated point	Definition 3.2.6
limit point	Definition 3.2.4
dense subset	P.95: G is dense in R if every element of R is a limit point of G
nowhere dense set	Definition 3.5.3
closure of a set	Definition 3.2.11
open set	Definition 3.2.1
closed set	Definition 3.2.7
compact set	Definition 3.3.1
bounded set	Definition 3.3.3
G-delta set	Definition 3.5.1
F-sigma set	Definition 3.5.1
open cover	Definition 3.3.6
finite subcover	Definition 3.3.6
function limit	Definition 4.2.1
continuous function	Definition 4.3.1
uniformly continuous function	Definition 4.4.5
right hand limit	Definition 4.6.2
left hand limit	Definition 4.6.2
removeable discontinuity	Exercise 4.6.3
jump discontinuity
essential discontinuity
monotone function	Definition 4.6.1
increasing function
decreasing function
alpha-continuous function	Definition 4.6.5
pointwise convergence	Definition 6.2.1B
uniform convergence	Definition 6.2.3
partition	Definition 7.2.1
upper sum
lower sum
refinement	Definition 7.2.2
upper integral	Definition 7.2.5
lower integral	Definition 7.2.5
Riemann-integrable	Definition 7.2.7
measure zero	Definition 7.6.1