arc
1:1
R ⊃ [a, b] t −→ x(t) = (x1(t), x1(t), . . . , xn(t)) ∈ Rn
A
x0
A
xi(t), i = 1, 2, . . . , n
x0 ∈ A
C2
C1
x0 ∈ A
∀ > 0 B (x0, )∩A = ∅
C3
ε
x
c1 : x(t) = (x(t), y(t)) = (1 − t, 1 − t) ,
c2 : x(t) = (x(t), y(t)) = (cos t, sin t) ,
f : Rn ⊃ A −→ R
t ∈ [0, 1]
π
t ∈ 0,
2
ε
x
u = f (x) ≡ f (x1 , x2 , . . . , xn )
u = u(t) = f (x1 (t), x2 (t), . . . , xn (t))
u = f (x, y) =
c2
xy
x2 +y 2
u = u(t) = cos t sin t
A
A
f (x)
A
x0 ∈ A
⎧
⎪
⎪
⎪
⎪
⎪
⎨
lim f (x) = L ⇔
x→x
0
⇔
∀ > 0, ∃ δ() > 0 :
x ∈ A, x − x0 < δ() |f (x) − L| < f (x) = f (x, y) = x4
x2 + y 2
lim
(x,y)→(0,0)
|f (x, y) − 0| = x4
x2
+
(x2
y2
y 2 )2
+
≤ =
x2 + y 2
⎪
⎪
⎩
f (x, y) = 0
x2 + y 2
3
|f (x) − 0| < ⎧
⎪
lim f (x(t))
⎪
⎪
⎪
t→b
⎪
⎨
x(t),
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
t ∈ (a, b) , lim x(t) = x0
⎪
⎪
⎪
t→b
⎪
⎩
x(t) ∈
⎪
⎪
⎪
⎪
⎪
⎭
< δ3 = 1
3
⎪
⎪
⎭
t→b
x(t) ∈
⇒
δ = ()
∀ > 0, ∃ δ() = > 0 :
x ∈ A, x − (0, 0) = x2 + y 2 < δ
x→x0
x(t),
⇒ lim f (x(t)) = L
⎪
⎪
t
∈
b]
,
lim
x(t)
=
x
[a,
t→b
⎪
⎪
0
⎪
⎪
A = R2 \(0, 0)
f (x, y)
⎫
⎪
⎪
⎪
⎪
⎪
⎬
lim f (x) = L
⇒
lim f (x, y)
x→x0
lim
(x,y)→(x0 ,y0 )
f (x, y) = lim
x→x0
= lim
y→y0
lim f (x, y)
y→y0
lim f (x, y)
x→x0
lim x(t) = x0
t→b
⎧
⎪
lim f (x) = L
⎪
⎪
x→x
⎪
0
⎪
⎨
{xk }k∈N ,
⎪
xk ∈ A, x0 ∈ A
⎪
⎪
⎪
⎪
⎩
lim xk = x0
k→∞
⇔
∀ ζ > 0 ∃ Δ(ζ) :
|t − b| < Δ(ζ) x(t) − x0 < ζ
lim f (x) = L ⇔
x→x
0
∀ > 0, ∃ δ() > 0 :
⇔
x ∈ A, x − x0 < δ() |f (x) − L| < ζ = δ()
∀ > 0, ∃ Z() = Δ(δ()) :
|t − b| < Z() x(t) − x0 < δ() |f (x) − L| < ⇒ lim f (x(t)) = L
t→b
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⇒
⎪
⎪
⎪
⎪
⎪
⎭
lim f (xk ) = L
k→∞
lim xk = x0
k→∞
⇔
∀ ζ > 0 ∃ N (ζ) :
k > N (ζ) xk − x0 < ζ
lim f (x) = L ⇔
x→x
0
∀ > 0, ∃ δ() > 0 :
⇔
x ∈ A, x − x0 < δ() |f (x) − L| < ζ = δ()
∀ > 0, ∃ M () = N (δ()) :
k > M () xk − x0 < δ() |f (xk ) − L| < ⇒ lim f (xk ) = L
t→k
⎧
⎪
⎪
⎪
⎨
A
⎪
⎪
⎪
⎩
x0 ∈ A
f (x)
f (x)
x0
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
⇔
⎧
⎪
⎨
⎧
⎪
lim f (xk )
⎪
⎪
⎨ k→∞
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎭
⇒
lim xk = x0
x0 ∈ A
k→∞
xk ∈ A,
∀ > 0, ∃ δ(, x0) > 0 :
⇔
x ∈ A,
⎪
⎩ x − x < δ(, x ) |f (x) − f (x )| < 0
0
0
⇒
⎫
⎪
⎬
⎪
⎭
A ⇔
A
lim f (x, y)
x→x0
f (x, y) = x2 + y 2
R2
⎧
⎪
⎨ f (x)
⎪
⎩
⇒ {f (x(t))
⎫
x(t),
x(t) ∈ A
A ⎪
⎬
⇒
, t ∈ [a, b]
⎪
⎭
∈ [a, b]}
⎧
⎪
⎨
⎫
⎪
⎬
A
⎪
⎩
f (x)
f (x)
⇔
⇔
∀ > 0, ∃ δ() > 0 : x
x0 ∈ A,
x − x0 < δ() |f (x) − f (x0)| < f (x, y) =
(x, y)
⎧
⎪
⎨
x2
1
+ y2 x2 + y 2 > 2
f (x)
⎫
⎪
⎬
f (x, y) =
R2
+
y2
x0
⇒
⎪
⎭
⇔
∃ > 0, ∀ δ > 0 : x
x0 ∈ A,
x − x0 < δ |f (x) − f (x0)| ≥ x2
⎪
⎪
⎪
⎪
⎪
⎩
⎫
⎪
{f (x)
A}
⎪
⎪
⎪
⎪
⎬
∀ > 0, x0 ∈ A, ∃δ(, x0) :
x − x0 < δ(, x0) f (x) − f (x0) < ⎪
⎪
⎪
⎪
⎪
inf δ(, x0) = Δ() = 0
⎭
f (x, y) =
A
⎪
⎩
⇔
A
⎪
⎭
⎧
⎪
⎪
⎪
⎪
⎪
⎨
1
x+y
f (x)
x > 1, y > 1
f (x)
g(t)
A,
t ∈ f (A)
⇒ {g ◦ f
Rn
⇒
A}
f
⊃ AM M / f (A) ⊂
g
MM
MM
M& g◦f
R
R
f (x)
g(t)
A,
t ∈ f (A)
⇒
⇒ {g ◦ f
Rn
A}
f
⊃ AM M / f (A) ⊂
g◦f
f (x, y) =
sin(xy)
xy
g
MM
MM
M& R
R
{f (x)
R2
∀ > 0 ∃ δ() :
lim g(t) = g(t0 ) |t − t0 | < δ() |g(t) − g(t0 )| < t→t0
∀ > 0, ∃ δ() > 0 : x
x0 ∈ A,
lim f (x) = f (x0 ) x − x0 < δ() |f (x) − f (x0 )| < x→x0
C} ⇒
f (C)
{f (x)
C} ⇒
⎧
⎫
⎪
⎪
∃
x
∈
C
:
f
(x
)
=
max
f
(x)
⎨
⎬
M
M
x∈C
⇒
⎪
⎩ ∃ xm ∈ C : f (xm) = min f (x) ⎪
⎭
x∈C
f (C)
{yk }k∈N
f (C)
sup f (C) ∈ f (C)
yk = f (xk ) −→ ∞
k→∞
C
xk
lim xk = x0 xk
yk = f (xk ) −→ f (x0)
k→∞
C
∃ sup f (C) ∀ k ∈ N ∃ yk = f (xk ) ∈ f (C) : sup f (C) −
1
k
< yk ≤
sup f (C)
lim f (xk ) = sup f (C) ∃ xk
k→∞
xk
lim xk = xM ∈ C
k→∞
f (x)
C
lim f (xk ) = f (xM ) =
k→∞
x∈C
{f (x)
k→∞
f (x)
∃ xM ∈ C : f (xM ) = max f (x)
C} ⇒
f (C)
⎧
⎪
⎨
{f (x)
C} ⇒
⎧
⎫
⎪
⎪
∃
x
∈
C
:
f
(x
)
=
max
f
(x)
⎨
⎬
M
M
x∈C
⇒
⎪
⎩ ∃ xm ∈ C : f (xm) = min f (x) ⎪
⎭
x∈C
{f (x)
C} ⇒
{f (x)
}
⎫
⎪
⎬
x1, x2 ∈ A
∃
x(t) ∈ A, t ∈ [t1, t2]
⎪
⎪
⎩
⎭
x(t1) = x1, x(t2) = x2
f (x)
A
A
⇒
x1 , x2 ∈ A
∃x ∈ A :
f (x1) < f (x2)
f (x1) < f (x) < f (x2)