## Advanced Calculus (II)

W^{EN}-C^{HING}L^{IEN}

Department of Mathematics National Cheng Kung University

2009

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

## Ch10: Metric Spaces

### 10.2: Limit Of Functions

Definition (10.22)

A point a is said to be a cluster point (of X ) if and only if Bδ(a) contains infinitely many points for each δ > 0.

Notation:

E is a subspace of X .

B_{r}^{E}(a) := {x ∈ E : ρ(x , a) < r }.(relative balls in E )

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

## Ch10: Metric Spaces

### 10.2: Limit Of Functions

Definition (10.22)

A point a is said to be a cluster point (of X ) if and only if Bδ(a) contains infinitely many points for each δ > 0.

Notation:

E is a subspace of X .

B_{r}^{E}(a) := {x ∈ E : ρ(x , a) < r }.(relative balls in E )

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

Example (10.24)

Show that every point in the interval [0,1] is a cluster point of the open interval (0,1).

Proof.

Let x0 ∈ [0, 1] and δ > 0. Then x_{0}+ δ >0 and x0− δ < 1.

In particular, (x0− δ, x_{0}+ δ) ∩ (0, 1) is itself a

nondegenerate interval, say (a, b). But (a, b) contains infinitely many points, e.g., (a + b)/2, (2a + b)/3, (3a + b)/4, . . .. Therefore, x0 is a cluster point of (0, 1).

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

Example (10.24)

Show that every point in the interval [0,1] is a cluster point of the open interval (0,1).

Proof.

Let x0 ∈ [0, 1] and δ > 0.Then x0+ δ >0 and x0− δ < 1.

In particular, (x0− δ, x_{0}+ δ) ∩ (0, 1) is itself a

nondegenerate interval, say (a, b).But (a, b) contains infinitely many points, e.g., (a + b)/2, (2a + b)/3, (3a + b)/4, . . .. Therefore, x0 is a cluster point of (0, 1).

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

Example (10.24)

Show that every point in the interval [0,1] is a cluster point of the open interval (0,1).

Proof.

Let x0 ∈ [0, 1] and δ > 0. Then x_{0}+ δ >0 and x0− δ < 1.

In particular, (x0− δ, x_{0}+ δ) ∩ (0, 1) is itself a

nondegenerate interval, say (a, b). But (a, b) contains infinitely many points,e.g., (a + b)/2, (2a + b)/3, (3a + b)/4, . . .. Therefore, x0 is a cluster point of (0, 1).

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

Example (10.24)

Show that every point in the interval [0,1] is a cluster point of the open interval (0,1).

Proof.

Let x0 ∈ [0, 1] and δ > 0. Then x_{0}+ δ >0 and x0− δ < 1.

In particular, (x0− δ, x_{0}+ δ) ∩ (0, 1) is itself a

nondegenerate interval, say (a, b).But (a, b) contains infinitely many points, e.g., (a + b)/2, (2a + b)/3, (3a + b)/4, . . .. Therefore, x0 is a cluster point of (0, 1).

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

Example (10.24)

Show that every point in the interval [0,1] is a cluster point of the open interval (0,1).

Proof.

Let x0 ∈ [0, 1] and δ > 0. Then x_{0}+ δ >0 and x0− δ < 1.

In particular, (x0− δ, x_{0}+ δ) ∩ (0, 1) is itself a

nondegenerate interval, say (a, b). But (a, b) contains infinitely many points,e.g., (a + b)/2, (2a + b)/3, (3a + b)/4, . . .. Therefore, x0 is a cluster point of (0, 1).

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

Example (10.24)

Show that every point in the interval [0,1] is a cluster point of the open interval (0,1).

Proof.

Let x0 ∈ [0, 1] and δ > 0. Then x_{0}+ δ >0 and x0− δ < 1.

In particular, (x0− δ, x_{0}+ δ) ∩ (0, 1) is itself a

nondegenerate interval, say (a, b). But (a, b) contains infinitely many points, e.g., (a + b)/2, (2a + b)/3, (3a + b)/4, . . .. Therefore, x0 is a cluster point of (0, 1).

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

Example (10.24)

Show that every point in the interval [0,1] is a cluster point of the open interval (0,1).

Proof.

Let x0 ∈ [0, 1] and δ > 0. Then x_{0}+ δ >0 and x0− δ < 1.

In particular, (x0− δ, x_{0}+ δ) ∩ (0, 1) is itself a

nondegenerate interval, say (a, b). But (a, b) contains infinitely many points, e.g., (a + b)/2, (2a + b)/3, (3a + b)/4, . . .. Therefore, x0 is a cluster point of (0, 1).

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

Definition (10.25)

Given (X , ρ), (Y , τ ), let a be a cluster point of X and f : X \{a} → Y . Then f (x ) is said to

converge to L, as x approaches a, if and only if for every ε >0 there is a δ > 0 such that

(1) 0 < ρ(x , a) < δ implies τ (f (x ), L) < ε.

In this case we write

L = lim

x →af (x )

and call L the limit of f (x ) as x approaches a.

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

Theorem (10.26)

Let a be a cluster point of X and f , g : X \{a} → Y . (i) If f (x ) = g(x ) for all x ∈ X \{a} and f (x ) has a limit as x → a, then g(x ) also has a limit as x → a and

x →alimg(x ) = lim

x →af (x ).

(ii) [Sequential Characterizations of Limits]. The limit L := lim

x →af (x )

exists if and only if f (xn) →L as n → ∞ for every sequence xn ∈ X \{a} that converges to a as n → ∞.

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

Theorem (10.26)

Let a be a cluster point of X and f , g : X \{a} → Y . (i) If f (x ) = g(x ) for all x ∈ X \{a} and f (x ) has a limit as x → a, then g(x ) also has a limit as x → a and

x →alimg(x ) = lim

x →af (x ).

(ii) [Sequential Characterizations of Limits]. The limit L := lim

x →af (x )

exists if and only if f (xn) →L as n → ∞ for every sequence xn ∈ X \{a} that converges to a as n → ∞.

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

Theorem (10.26)

Let a be a cluster point of X and f , g : X \{a} → Y . (i) If f (x ) = g(x ) for all x ∈ X \{a} and f (x ) has a limit as x → a, then g(x ) also has a limit as x → a and

x →alimg(x ) = lim

x →af (x ).

(ii) [Sequential Characterizations of Limits]. The limit L := lim

x →af (x )

exists if and only if f (xn) →L as n → ∞ for every sequence xn ∈ X \{a} that converges to a as n → ∞.

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

Theorem (10.26)

(iii) Suppose that Y =**R**^{n}. If f (x ) and g(x ) have a limit as
x approaches a, then so do (f + g)(x ), (f · g)(x ), (αf )(x ),
and (f /g)(x ) (when Y =**R and the limit of g(x ) is**

nonzero). In fact,

x →alim(f + g)(x ) = lim

x →af (x ) + lim

x →ag(x ),

x →alim(αf )(x ) = α lim

x →af (x ),

x →alim(f · g)(x ) = lim

x →af (x ) · lim

x →ag(x ),
and (when Y =**R and the limit of g(x ) is nonzero)**

x →alim

f g

(x ) = limx →af (x ) limx →ag(x ).

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

Theorem (10.26)

(iv) [Squeeze Theorem for Functions] Suppose that
Y =**R. If h : X \{a} → R satisfies g(x ) ≤ h(x ) ≤ f (x ) for**
all x ∈ X \{a}, and

x →alimf (x ) = lim

x →ag(x ) = L, then the limit of h exists, as x → a, and

x →alimh(x ) = L.

(v) [Comparison Theorem for Functions] Suppose that
Y =**R. If f (x ) ≤ g(x ) for all x ∈ X \{a}, and f and g have**
a limit as x approaches a, then

x →alimf (x ) ≤ lim

x →ag(x ).

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

Theorem (10.26)

(iv) [Squeeze Theorem for Functions] Suppose that
Y =**R. If h : X \{a} → R satisfies g(x ) ≤ h(x ) ≤ f (x ) for**
all x ∈ X \{a}, and

x →alimf (x ) = lim

x →ag(x ) = L, then the limit of h exists, as x → a, and

x →alimh(x ) = L.

(v) [Comparison Theorem for Functions] Suppose that
Y =**R. If f (x ) ≤ g(x ) for all x ∈ X \{a}, and f and g have**
a limit as x approaches a, then

x →alimf (x ) ≤ lim

x →ag(x ).

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

Definition (10.27)

Let E be a nonempty subset of X and f : E → Y .

(i) f is said to be continuous at a point a ∈ E if and only if given ε there is a δ > 0 such that

ρ(x , a) < δ and x ∈ E imply τ (f (x ), f (a)) < ε.

(ii) f is said to be continuous on E (Notation: f : E → Y is continuous) if and only if f is continuous at every x ∈ E .

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

Theorem (10.28)

Let E be a nonempty subset of X and f , g : E → Y . (i) f is continuous at a ∈ E if and only if f (xn) →f (a), as n → ∞, for all sequences xn∈ E that converge to a.

(ii) Suppose that Y =**R**^{n}. If f , g are continuous at a point
a ∈ E (respectively, continuous on a set E), then so are
f + g, f · g, and αf (for any α ∈**R). Moreover, in the case**
Y =**R, f /g is continuous at a ∈ E when g(a) 6= 0**

(respectively, on E when g(x ) 6= 0 for all a ∈ E .)

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

Theorem (10.28)

Let E be a nonempty subset of X and f , g : E → Y . (i) f is continuous at a ∈ E if and only if f (xn) →f (a), as n → ∞, for all sequences xn∈ E that converge to a.

(ii) Suppose that Y =**R**^{n}. If f , g are continuous at a point
a ∈ E (respectively, continuous on a set E), then so are
f + g, f · g, and αf (for any α ∈**R). Moreover, in the case**
Y =**R, f /g is continuous at a ∈ E when g(a) 6= 0**

(respectively, on E when g(x ) 6= 0 for all a ∈ E .)

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

Theorem (10.28)

Let E be a nonempty subset of X and f , g : E → Y . (i) f is continuous at a ∈ E if and only if f (xn) →f (a), as n → ∞, for all sequences xn∈ E that converge to a.

(ii) Suppose that Y =**R**^{n}. If f , g are continuous at a point
a ∈ E (respectively, continuous on a set E), then so are
f + g, f · g, and αf (for any α ∈**R). Moreover, in the case**
Y =**R, f /g is continuous at a ∈ E when g(a) 6= 0**

(respectively, on E when g(x ) 6= 0 for all a ∈ E .)

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

Theorem (10.29)

Suppose that X , Y , and Z are metric spaces, a is a cluster point of X , f : X → Y , and g : f (X ) → Z . If f (x ) → L as x → a and g is continuous at L, then

x →alim(g ◦ f )(x ) = g

x →alim(f )(x )

.

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

## Thank you.

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