## Advanced Calculus (I)

WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

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

## 3.4 Uniform continuity

Definition

* Let E be a nonempty subset of R and f* :

*E*→

**R.**Then f is

*said to be uniformly continuous on E (notation: f*:

*E*→

**R**is uniformly continuous) if and only if for every ǫ >0 there is aδ >0 such that

|x −*a| < δ* *and x,a*∈*E imply* |f(x) −*f*(a)| < ǫ.

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

## 3.4 Uniform continuity

Definition

* Let E be a nonempty subset of R and f* :

*E*→

**R.**Then f is

*said to be uniformly continuous on E (notation: f*:

*E*→

**R**is uniformly continuous) if and only if for every ǫ >0 there is aδ >0 such that

|x −*a| < δ* *and x,a*∈*E imply* |f(x) −*f*(a)| < ǫ.

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

**Example:**

*Show that f*(x) =*x*^{2}**is not uniformly continuous on R.**

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

**Example:**

*Show that f*(x) =*x*^{2}**is not uniformly continuous on R.**

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

**Lemma:**

*Suppose that E* ⊆*R and f* :*E* →**R is uniformly**
*continuous. If x**n*∈*E is Cauchy, then f*(x*n*)is Cauchy.

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

**Lemma:**

*Suppose that E* ⊆*R and f* :*E* →**R is uniformly**
*continuous. If x**n*∈*E is Cauchy, then f*(x*n*)is Cauchy.

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

Theorem

*Suppose that I is a closed, bounded interval. If f* :*I* →**R**
*is continuous on I, then f is uniformly continuous on I.*

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

Theorem

*Suppose that I is a closed, bounded interval. If f* :*I* →**R**
*is continuous on I, then f is uniformly continuous on I.*

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

**Proof:**

Suppose to the contrary that f is continuous but not
uniformly continuous on I Then there is anǫ0 >0 and
*points x**n*,*y**n*∈*I* such that|x*n*−*y**n*| <1/n and

(10) |f(x*n*) −*f*(y*n*)| ≥ ǫ0, *n*∈**N.**

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

**Proof:**

Suppose to the contrary that f is continuous but not
uniformly continuous on I Then there is anǫ0 >0 and
*points x**n*,*y**n*∈*I such that*|x*n*−*y**n*| <1/n and

(10) |f(x*n*) −*f*(y*n*)| ≥ ǫ0, *n*∈**N.**

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

**Proof:**

Suppose to the contrary that f is continuous but not
uniformly continuous on I Then there is anǫ0 >0 and
*points x**n*,*y**n*∈*I* such that|x*n*−*y**n*| <1/n and

(10) |f(x*n*) −*f*(y*n*)| ≥ ǫ0, *n*∈**N.**

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

**Proof:**

Suppose to the contrary that f is continuous but not
uniformly continuous on I Then there is anǫ0 >0 and
*points x**n*,*y**n*∈*I such that*|x*n*−*y**n*| <1/n and

(10) |f(x*n*) −*f*(y*n*)| ≥ ǫ0, *n*∈**N.**

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

By the Bolzano-Weierstrass Theorem and the
Comparison Theorem, the sequence {x*n*}has a
subsequence, *say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly, the sequence*{y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y), a contradiction. 2

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

By the Bolzano-Weierstrass Theorem and the
Comparison Theorem, the sequence{x*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I.* Similarly, the sequence{y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y), a contradiction. 2

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

By the Bolzano-Weierstrass Theorem and the
Comparison Theorem, the sequence {x*n*}has a
subsequence, *say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly,*the sequence {y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y), a contradiction. 2

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

By the Bolzano-Weierstrass Theorem and the
Comparison Theorem, the sequence {x*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I.* Similarly, the sequence{y*n** _{k}*}

*k*∈Nhas a convergent subsequence,

*say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y), a contradiction. 2

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

By the Bolzano-Weierstrass Theorem and the
Comparison Theorem, the sequence {x*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly,*the sequence {y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I.*

*Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y), a contradiction. 2

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

*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly, the sequence*{y*n** _{k}*}

*k*∈Nhas a convergent subsequence,

*say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y), a contradiction. 2

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

By the Bolzano-Weierstrass Theorem and the
Comparison Theorem, the sequence {x*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly, the sequence*{y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I.*

*Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0;i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y), a contradiction. 2

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

*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly, the sequence*{y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y), a contradiction. 2

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

By the Bolzano-Weierstrass Theorem and the
Comparison Theorem, the sequence {x*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly, the sequence*{y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0;i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N,**so Theorem

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y), a contradiction. 2

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

*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly, the sequence*{y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y .*Therefore,

*f*(x) =

*f*(y), a contradiction. 2

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

By the Bolzano-Weierstrass Theorem and the
Comparison Theorem, the sequence {x*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly, the sequence*{y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N,**so Theorem

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y),a contradiction. 2

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

*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly, the sequence*{y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y .*Therefore,

*f*(x) =

*f*(y), a contradiction. 2

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

By the Bolzano-Weierstrass Theorem and the
Comparison Theorem, the sequence {x*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly, the sequence*{y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y),a contradiction. 2

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

*n*}has a
*subsequence, say x**n**k**, that converges, as k* → ∞, to
*some x* ∈*I. Similarly, the sequence*{y*n** _{k}*}

*k*∈Nhas a

*convergent subsequence, say y*

*n*

*, that converges, as*

_{kj}*j*→ ∞, to some y ∈

*I. Since x*

*n*

*→*

_{kj}*x as j*→ ∞and f is continuous, it follows from (10) that|f(x) −

*f*(y)| ≥ ǫ0; i.e.,

*f*(x) 6=

*f*(y). But|x

*n*−

*y*

*n*| <1/n for all n∈

**N, so Theorem**

*2.9 (the Squeeze Theorem) implies that x*=

*y . Therefore,*

*f*(x) =

*f*(y), a contradiction. 2

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

Theorem

*Let*(a,*b)* *be a bounded, open, nonempty interval and*
*f* : (a,*b) → R. Then f is uniformly continuous on*(a,

*b)if*

*and only if f can be extended continuously to*[a,

*b], i.e., if*

*and only if there is a continuous function g*: [a,

*b] →*

**R**

*that satifies*

(11) *f*(x) =*g(x*), *x* ∈ (a,*b).*

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

Theorem

*Let*(a,*b)* *be a bounded, open, nonempty interval and*
*f* : (a,*b) → R. Then f is uniformly continuous on*(a,

*b)if*

*and only if f can be extended continuously to*[a,

*b], i.e., if*

*and only if there is a continuous function g*: [a,

*b] →*

**R**

*that satifies*

(11) *f*(x) =*g(x*), *x* ∈ (a,*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:**

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)**

*Thank you.*

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