Chapter 4 (part a)

Question 1

Functional Limit- Let f: A to R and let c be a limit point of a domain A. We say that limit x approaches c of f(x) = L provided that….

Answer 1

for all ε > 0, there exists a δ > 0 such that whenever

0 < |x - c| < δ

it follows that

|f(x) - L| < ε

Question 2

for all ε > 0, there exists a δ > 0 such that whenever

0 < |x - c| < δ

it follows that

|f(x) - L| < ε

Answer 2

Functional Limit- Let f: A to R and let c be a limit point of a domain A. We say that limit x approaches c of f(x) = L provided that….

Question 3

Functional Limit (Topological Version)-

Let c be a limit point of the domain of the domain of

f:A to R.

Then, the limit as x approaches c of f(x) = L if….

Answer 3

for every ε-neighborhood V_ε(L) of L, there exists a δ-neighborhood V_δ(c) around c with the property that for all x in V_δ(c) different from c if follows that

f(x) in V_ε(L).

Question 4

for every ε-neighborhood V_ε(L) of L, there exists a δ-neighborhood V_δ(c) around c with the property that for all x in V_δ(c) different from c if follows that

f(x) in V_ε(L).

Answer 4

Functional Limit (Topological Version)-

Let c be a limit point of the domain of the domain of

f:A to R.

Then, the limit as x approaches c of f(x) = L if….

Question 5

Sequential Criterion for Functional Limits-

Given a function f: A to R and a limit point c of A, the following two statements are equivalent:

Answer 5

i. ) lim x approaches c of f(x) = L.
ii. ) For all sequences (x_n) c A satisfying x_n not equal to c and limit (x_n) goes to c, it follows that limit f(x_n) goes to L.

Question 6

i. ) lim x approaches c of f(x) = L.
ii. ) For all sequences (x_n) c A satisfying x_n not equal to c and limit (x_n) goes to c, it follows that limit f(x_n) goes to L.

Answer 6

Sequential Criterion for Functional Limits-

Given a function f: A to R and a limit point c of A, the following two statements are equivalent:

Question 7

Algebraic Limit Theorem for Functional Limits-

Let f and g be functions defined on a domain A c R, and assume lim x approaches c of f(x) = L and lim x approaches c of g(x) = M for some limit point c of A. Then,

Answer 7

i. ) lim x to c of k*f(x) = k*L for k in R
ii. ) lim x to c of [f(x) + g(x)] = L + M
iii. ) lim x to c of f(x)*g(x) = L*M
iv. ) lim x to c of f(x) / g(x) = L/M for M not 0.

Question 8

i. ) lim x to c of k*f(x) = k*L for k in R
ii. ) lim x to c of [f(x) + g(x)] = L + M
iii. ) lim x to c of f(x)*g(x) = L*M
iv. ) lim x to c of f(x) / g(x) = L/M for M not 0.

Answer 8

Algebraic Limit Theorem for Functional Limits-

Let f and g be functions defined on a domain A c R, and assume lim x approaches c of f(x) = L and lim x approaches c of g(x) = M for some limit point c of A. Then,

Question 9

Divergence Criterion for Functional Limits-

Let f be a function defined on A, and let c be a limit point of A. If there exists two sequences (x_n) and (y_n) in A with x_n not equal to c and y_n not equal to c…

Answer 9

and lim x_n = lim y_n = c

but lim f(x_n) does not equal the lim f(y_n),

then the functional limit as x approaches c of f(x) does not exist.

Question 10

and lim x_n = lim y_n = c

but lim f(x_n) does not equal the lim f(y_n),

then the functional limit as x approaches c of f(x) does not exist.

Answer 10

Divergence Criterion for Functional Limits-

Let f be a function defined on A, and let c be a limit point of A. If there exists two sequences (x_n) and (y_n) in A with x_n not equal to c and y_n not equal to c…

Question 11

Continuity- A function f: A to R is continuous at a point c in A if…

Answer 11

for all ε > 0, there exists a δ > 0 such that whenever

|x - c| < δ

it follows that

|f(x) - f(c)| < ε

Question 12

for all ε > 0, there exists a δ > 0 such that whenever

|x - c| < δ

it follows that

|f(x) - f(c)| < ε

Answer 12

Continuity- A function f: A to R is continuous at a point c in A if…

Question 13

Characterizations of Continuity-

Let f: A to R, and let c be in A. The function f is continuous at c iff one of the following three conditions is met:

Answer 13

i.) for all ε > 0, there exists a δ > 0 such that

|x - c| < δ and |f(x) - f(c)| < ε

ii. ) For all V_ε(f(c)), there exists a V_δ(c) with the property that x in V_δ(c) implies f(x) in V_ε(f(c))
iii. ) I (x_n) goes to c, then f(x_n) goes to f(c)

Question 14

i.) for all ε > 0, there exists a δ > 0 such that

|x - c| < δ and |f(x) - f(c)| < ε

ii. ) For all V_ε(f(c)), there exists a V_δ(c) with the property that x in V_δ(c) implies f(x) in V_ε(f(c))
iii. ) I (x_n) goes to c, then f(x_n) goes to f(c)

Answer 14

Characterizations of Continuity-

Let f: A to R, and let c be in A. The function f is continuous at c iff one of the following three conditions is met:

Question 15

Criterion for Discontinuity-

Let f: A to R, and let c in A be a limit point of A. If there exists a sequence…

Answer 15

(x_n) c A where (x_n) goes to c but such that f(x_n) does not converge to f(c), we conclude that f is not continuous at c.

Question 16

(x_n) c A where (x_n) goes to c but such that f(x_n) does not converge to f(c), we conclude that f is not continuous at c.

Answer 16

Criterion for Discontinuity-

Let f: A to R, and let c in A be a limit point of A. If there exists a sequence…

Question 17

Algebraic Continuity Theorem-

Assume f: A to R an g: A to R are continuous at a point c in A. Then…

Answer 17

i. ) k*f(x) is continuous at c for k in R
ii. ) f(x) + g(x) is continuous at c
iii. ) f(x)*g(x) is continuous at c
iv. ) f(x) / g(x) is continuous at c for g(x) not 0

Question 18

i. ) k*f(x) is continuous at c for k in R
ii. ) f(x) + g(x) is continuous at c
iii. ) f(x)*g(x) is continuous at c
iv. ) f(x) / g(x) is continuous at c for g(x) not 0

Answer 18

Algebraic Continuity Theorem-

Assume f: A to R an g: A to R are continuous at a point c in A. Then…

Question 19

Composition of Continuous Functions-

Given f: A to R and g: B to R, assume that the range

f(A) = {f(x): x in A}

is contained in the domain B so that the composition g(f(x)) is defined on A.

If f is continuous at c in A, and g is continuous at f(c) in B, then…

Answer 19

g(f(x)) is continuous at c.

Question 20

g(f(x)) is continuous at c.

Answer 20

Composition of Continuous Functions-

Given f: A to R and g: B to R, assume that the range

f(A) = {f(x): x in A}

is contained in the domain B so that the composition g(f(x)) is defined on A.

If f is continuous at c in A, and g is continuous at f(c) in B, then…