9. 写像の連続性

9.

2007-06-14

9.1

,

,

R ²

B = {

((1 + e ^{− t} ) cos t, (1 + e ^{− t} ) sin t) | t ∈ R }

の開核

,

,

,

,

,

.

9.2

–

,

R

R

f (x) = sin x

,

U = (0, π )

, f (U ), f ^{− 1} (U )

.

9.3

?

[A] x _n =

 



100 (n < 100) 1/n (n ≤ 100)

[B] x _n = ^{( −} _n ¹⁾

.

[C] x _2n = _n ¹ , x _2n+1 = 1.

[D] x _2n = _n ¹ , x _2n+1 = 0.

9.4

?

f (x) =

 

 

 



+1 x > 0 0 x = 0

− 1 x < 0 .

x _n = _n ¹ .

n lim →∞ f (x _n ) =?

[A] +1 [B] − 1 [C] 0

[D]

.

9.5

?

f (x) =

 

 

 



+1 x > 0 0 x = 0

− 1 x < 0 .

x _n = ^{( −} _n ¹⁾

.

n lim →∞ f (x _n ) =?

[A] +1 [B] − 1 [C] 0

[D]

.

9.6

?

f (x) =

 

 

 



+1 x > 0 0 x = 0

− 1 x < 0 .

x _2n = ( − 1) ⁿ _n ¹ , x _2n+1 = 0.

n lim →∞ f (x _n ) =?

[A] +1 [B] − 1 [C] 0

[D]

.