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# Proof:

Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;

hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim

n→∞f(xn)

exists. This value does not change if we use a different sequence to a approximate b.

WEN-CHINGLIEN Advanced Calculus (I)

### Proof:

Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;

hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim

n→∞f(xn)

exists. This value does not change if we use a different sequence to a approximate b.

WEN-CHINGLIEN Advanced Calculus (I)

### Proof:

Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;

hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim

n→∞f(xn)

exists. This value does not change if we use a different sequence to a approximate b.

WEN-CHINGLIEN Advanced Calculus (I)

### Proof:

Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;

hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim

n→∞f(xn)

exists. This value does not change if we use a different sequence to a approximate b.

WEN-CHINGLIEN Advanced Calculus (I)

### Proof:

Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;

hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim

n→∞f(xn)

exists. This value does not change if we use a different sequence to a approximate b.

WEN-CHINGLIEN Advanced Calculus (I)

### Proof:

Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;

hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim

n→∞f(xn)

exists. This value does not change if we use a different sequence to a approximate b.

WEN-CHINGLIEN Advanced Calculus (I)

### Proof:

Suppose that f is uniformly continuous on(a,b). Let xn ∈ (a,b)converge to b as n→ ∞. Then{xn}is Cauchy;

hence, by Lemma 3.38, so is{f(xn)}. In particular, g(b) := lim

n→∞f(xn)

exists. This value does not change if we use a different sequence to a approximate b.

WEN-CHINGLIEN Advanced Calculus (I)

Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Givenǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞, we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Indeed,let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0,chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞, we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Givenǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞, we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0,chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞, we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞, we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞, we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞,we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞, we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞,we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞, we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Indeed, let yn ∈ (a,b)be another sequence that converges to b as n→ ∞. Given ǫ >0, chooseδ >0 such that (9) holds for E = (a,b). Since xnyn →0, choose NN so that nN implies|xnyn| < δ. By (9), then,|f(xn) −f(yn)| < ǫfor all nN. Taking the limit of this inequality as n → ∞, we obtain

| lim

n→∞f(xn) − lim

n→∞f(yn)| ≤ ǫ

for all ǫ >0. It follows from Theorem 1.9 that

n→∞lim f(x)= lim

n→∞f(yn).

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus,f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus,f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b];hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b];hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

Thus, g(b)is well defined. A similar argument defines g(a). Set g(x) =f(x)for x ∈ (a,b). Then g is defined on [a,b], satifies (11), and is continuous on[a,b]by the Sequential Characterization of Limits. Thus, f can be

“continuously extended” to g as required.

Conversely, suppose that there is a function g

continuouos on [a,b] that satifies (11). By theorem 3.39, g is uniformly continuous on [a,b]; hence, g is uniformly continuous on (a,b). We conclude that f is uniformly continuous on (a,b).2

WEN-CHINGLIEN Advanced Calculus (I)

### Example:

Prove that f(x) = (x −1)

log x is uniformly continuous on (0,1).

WEN-CHINGLIEN Advanced Calculus (I)

### Example:

Prove that f(x) = (x −1)

log x is uniformly continuous on (0,1).

WEN-CHINGLIEN Advanced Calculus (I)