MA362 Midterm Study Guide

The midterm examination will be held Thursday, March 13th.

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 that continuous functions preserve compactness (Theorem 4.4.2)
- Prove that continuous functions preserve connectedness (Theorem 4.5.2)
- Prove that a function continuous on a compact set K is uniformly continuous on K(Theorem 4.4.8)
- Prove that a functional limit at a point exists if and only if the left and right hand functional limits exist.
Two proofs you have not seen before, chosen from a list of four possibilities that will be supplied.

Proofs for Theorems 4.4.2 and 4.5.2 are given in the text. A proof of Theorem 4.4.8 also appears in the text, and a second proof is posted on this site, as is a proof of the fourth proposition in the list. These are also included in the .

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	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