Chapter 4 (part b)

Question 1

Preservation of Compact Sets-

Let f: A to R be continuous on A. If K c A is compact, then…

Answer 1

f(K) is compact as well.

Question 2

f(K) is compact as well.

Answer 2

Preservation of Compact Sets-

Let f: A to R be continuous on A. If K c A is compact, then…

Question 3

Extreme Value Theorem-

If f: K to R is continuous on a compact set K c R, then…

Answer 3

f attains a maximum and minimum value. In other words, there exists x₀, x₁ in K such that

f(x₀) < f(x) < f(x₁) for all x in K.

Question 4

f attains a maximum and minimum value. In other words, there exists x₀, x₁ in K such that

f(x₀) < f(x) < f(x₁​) for all x in K.

Answer 4

Extreme Value Theorem-

If f: K to R is continuous on a compact set K c R, then…

Question 5

Uniform Continuity-

A function f: A to R is uniformly continuous on A if…

Answer 5

for every ε > 0 there exists a δ > 0 such that for all x,y in A

|x - y| < δ

implies |f(x) - f(y)| < ε

Question 6

for every ε > 0 there exists a δ > 0 such that for all x,y in A

|x - y| < δ

implies |f(x) - f(y)| < ε

Answer 6

Uniform Continuity-

A function f: A to R is uniformly continuous on A if…

Question 7

Sequential Criterion for Absence of Uniform Continuity-

A function f: A to R fails to be uniformly continuous on A if and only if…

Answer 7

there exists a particular ε₀ > 0 and two sequences (x_n) and (y_n) in A satisfying

|x_n - y_n| goes to 0

but |f(x_n) - f(y_n)| > ε₀

Question 8

there exists a particular ε₀ > 0 and two sequences (x_n) and (y_n) in A satisfying

|x_n - y_n| goes to 0

but |f(x_n) - f(y_n)| > ε₀

Answer 8

Sequential Criterion for Absence of Uniform Continuity-

A function f: A to R fails to be uniformly continuous on A if and only if…

Question 9

Uniform Continuity on Compat Sets-

A function tha is continuous on a compact set K…

Answer 9

is uniformly continuous on K.

Question 10

is uniformly continuous on K.

Answer 10

Uniform Continuity on Compat Sets-

A function tha is continuous on a compact set K…

Question 11

Intermediate Value Theorem-

Let f: [a,b] to R be continuous. If L is a real number satisfying f(a) < L < f(b), then…

Answer 11

there exists a point c in (a,b) where f(c) = L.

Question 12

there exists a point c in (a,b) where f(c) = L.

Answer 12

Intermediate Value Theorem-

Let f: [a,b] to R be continuous. If L is a real number satisfying f(a) < L < f(b), then…

Question 13

Preservation of Connected Sets-

Let f: G to R be continuous. If E c G is connected…

Answer 13

then f(E) is connected as well.

Question 14

then f(E) is connected as well.

Answer 14

Preservation of Connected Sets-

Let f: G to R be continuous. If E c G is connected…

Question 15

Intermediate Value Property-

A function f has the intermediate value propery on an interval [a,b] if…

Answer 15

for all x < y in [a,b] and all L between f(x) and f(y), it is always possible to find a point c in (x,y) where f(c) = L.

Question 16

for all x < y in [a,b] and all L between f(x) and f(y), it is always possible to find a point c in (x,y) where f(c) = L.

Answer 16

Intermediate Value Property-

A function f has the intermediate value propery on an interval [a,b] if…