шамин с сдо
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10 |
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5 |
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f(x) |
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0 |
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−5 |
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−10 |
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−4 |
−2 |
0 |
2 |
4 |
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x |
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f(x) |
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N
RN
x = (x1, x2, . . . , xN ),
xi
x
x, y RN
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xi = yi, i = 1, . . . , N. |
x RN |
λ RN |
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λ · x = (λx1, . . . , λxN ). |
x, y RN
x ± y = (x1 ± y1, . . . , xN ± yN ).
x + y = y + x, (x + y) + z = x + (y + z), λ · (x + y) = λ · x + λ · y.
(0, 0, . . . , 0) 0
x, y RN
p
̺(x, y) = (x1 − y1)2 + · · · + (xN − yN )2.
̺(x, y) ≥ 0, ̺(x, y) = 0 x = y;
̺(x, y) = ̺(y, x);
̺(x, y) ≤ ̺(x, z) + ̺(z, y)
RN |
RN |
x0 |
r > 0 |
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Br(x0) = {x RN : ̺(x0, x) < r}. |
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N = 1 x0 = 5 r = 2
Br(x0) = (3, 7).
N = 2 x0 = 0 r = 1
q
Br(x0) = (x1, x2) : x21 + x22 < 1
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A RN |
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x0 RN |
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ε > 0 |
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Bε(x0) RN . |
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R2 |
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x′ |
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B1(0) |
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x = (x1′ , . . . , xN′ ) |
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r = ̺(x′, 0). |
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r < 1 |
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ε = |
1−r |
Br(x′) |
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2 |
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B1(0) |
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x′′ |
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Br(x′) |
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̺(x′′, 0) < ε + r < |
1 − r |
+ r < |
r + 1 |
< 1. |
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2 |
2 |
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x0
U(x0) |
x0 |
x0 U(x0).
M RN
R > 0
M BR(0).
Q Rn
RN
RN
RN
x1, x2, x3, . . . .
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{xn} RN |
x∞ |
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ε > 0 |
N = N(ε) |
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ε |
̺(xn, x∞) < ε, |
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n > N(ε) |
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xn → x∞, n → ∞ |
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lim x |
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= x . |
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n |
→∞ |
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∞ |
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x∞ RN |
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{xn} |
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U(x0) |
N = N(U(x0)) |
n > N(U(x0)) |
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xn U(x0).
N = 1
R1
N ≥ 2
±∞
R2
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xn = (x1n, x2n), |
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x1n = |
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cos n |
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= |
sin n |
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n |
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n |
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xn → 0 |
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n → ∞ |
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ε > 0 |
N(ε) > 1 |
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n > N(ε) |
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ε |
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s
̺(xn, 0) =
RN
xn → x0 n → ∞
A
cos2 n |
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sin2 n |
1 |
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+ |
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= |
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< ε. |
n2 |
n2 |
n |
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A RN |
x0 |
xn A,
x0
B1(0) R2
x0 = |
√ |
, √ |
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1 |
1 |
22
xn = |
√2 − |
1 + n |
, √2 − |
1 + n |
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1 |
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1 |
1 |
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1 |
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B1(0) |
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x0 |
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x0 |
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B1(0) |
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Q
π
{xn}
xn → π, n → ∞.
A RN
BR(x0)
BR(x0) = {x RN : ̺(x, x0) ≤ R},
x′ / BR(x0)
x′
R
R′ = ̺(x′, x0).
B |
R(x0) |
R′ > |
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Bε(x′) |
ε < R′ − R |
BR(x0) ∩ Bε(x′) = .
{xn}
B |
R(x0) |
xn → x′ |
A RN
A
A
R
Q
A = {x RN : 0 < ̺(x, 0) < 1}
B1(0)
(0, 1]
RN
RN
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A |
∂A |
x′ ∂A |
r > 0 |
Br(x′) |
A |
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A |
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BR(x0)
SR(x0) = {x RN : ̺(x, x0) = R}.
SR(x0) |
R |
x0
R = 1 x0 = 0
x2 
x′
Br(x′)
x1
B1(0)
B1(0)
D RN
D RN
f : D → R.