Advanced Calculus (II)
WEN-CHINGLIEN
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 .
BrE(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 .
BrE(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 x0+ δ >0 and x0− δ < 1.
In particular, (x0− δ, x0+ 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, x0+ 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, x0+ 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, x0+ 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, x0+ 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, x0+ 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, x0+ 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 =Rn. 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 =Rn. 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 =Rn. 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 =Rn. 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)