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

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

*n→∞**f*(x*n*)

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
*x**n* ∈ (a,*b)converge to b as n*→ ∞. Then{x*n*}is Cauchy;

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

*n→∞**f*(x*n*)

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*
*x**n* ∈ (a,*b)converge to b as n*→ ∞. Then{x*n*}is Cauchy;

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

*n→∞**f*(x*n*)

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

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*b). Let*
*x**n* ∈ (a,*b)converge to b as n*→ ∞. Then{x*n*}is Cauchy;

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

*n→∞**f*(x*n*)

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

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*b). Let*
*x**n* ∈ (a,*b)converge to b as n*→ ∞. Then{x*n*}is Cauchy;

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

*n→∞**f*(x*n*)

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

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*b). Let*
*x**n* ∈ (a,*b)converge to b as n*→ ∞. Then{x*n*}is Cauchy;

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

*n→∞**f*(x*n*)

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

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*b). Let*
*x**n* ∈ (a,*b)converge to b as n*→ ∞. Then{x*n*}is Cauchy;

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

*n→∞**f*(x*n*)

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

WEN-CHINGLIEN **Advanced Calculus (I)**

*Indeed, let y**n* ∈ (a,*b)*be another sequence that
*converges to b as n*→ ∞. Givenǫ >0, chooseδ >0
*such that (9) holds for E* = (a,*b). Since x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N. Taking the limit of*

*this inequality as n*→ ∞, we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

WEN-CHINGLIEN **Advanced Calculus (I)**

Indeed,*let y**n* ∈ (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 x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N. Taking the limit of*

*this inequality as n*→ ∞, we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

WEN-CHINGLIEN **Advanced Calculus (I)**

*Indeed, let y**n* ∈ (a,*b)*be another sequence that
*converges to b as n*→ ∞. Givenǫ >0, chooseδ >0
*such that (9) holds for E* = (a,*b).* *Since x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N. Taking the limit of*

*this inequality as n*→ ∞, we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

WEN-CHINGLIEN **Advanced Calculus (I)**

*Indeed, let y**n* ∈ (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 x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N. Taking the limit of*

*this inequality as n*→ ∞, we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

WEN-CHINGLIEN **Advanced Calculus (I)**

*Indeed, let y**n* ∈ (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 x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N. Taking the limit of*

*this inequality as n*→ ∞, we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

WEN-CHINGLIEN **Advanced Calculus (I)**

*Indeed, let y**n* ∈ (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 x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N.*Taking the limit of

*this inequality as n*→ ∞, we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

WEN-CHINGLIEN **Advanced Calculus (I)**

*Indeed, let y**n* ∈ (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 x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N. Taking the limit of*

*this inequality as n*→ ∞,we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

WEN-CHINGLIEN **Advanced Calculus (I)**

*Indeed, let y**n* ∈ (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 x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N.*Taking the limit of

*this inequality as n*→ ∞, we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

WEN-CHINGLIEN **Advanced Calculus (I)**

*Indeed, let y**n* ∈ (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 x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N. Taking the limit of*

*this inequality as n*→ ∞,we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

WEN-CHINGLIEN **Advanced Calculus (I)**

*Indeed, let y**n* ∈ (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 x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N. Taking the limit of*

*this inequality as n*→ ∞, we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

WEN-CHINGLIEN **Advanced Calculus (I)**

*Indeed, let y**n* ∈ (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 x**n*−*y**n* →0,
*choose N* ∈* N so that n* ≥

*N implies*|x

*n*−

*y*

*n*| < δ. By (9), then,|f(x

*n*) −

*f*(y

*n*)| < ǫ

*for all n*≥

*N. Taking the limit of*

*this inequality as n*→ ∞, we obtain

| lim

*n*→∞*f*(x*n*) − lim

*n*→∞*f*(y*n*)| ≤ ǫ

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

*n→∞*lim *f*(x)= lim

*n→∞**f*(y*n*).

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

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

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

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

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

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

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

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

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)**