Re: [考古] 一致連續的問題
※ 引述《s920911 (旋轉布丁)》之銘言：
: f(x)=1+x+x^2+x^3+... 下列區間和者不為一致連續?
: (A) 0≦x≦0.5
: (B) -0.9999<x<0.9999
: (C) 0<x<0.9999
: (D) 0<x<0.5
: (E) 0<x<1
: 答案 是E
: 我想問問 B的答案不行嗎?
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這裡的一致連續指的應該是 uniformly continuous
與一般的連續差別在於 δ 的選取只跟 ε 有關而跟點的選取無關
f(x) 只定義在 (-1,1) 且
1
f(x) = --------- .
1 - x
可以考慮以下這種點 x_n := 1-1/n , y_n := 1-1/(2n)
1
| f(x_n) - f(y_n) | = ------------------ |x_n-y_n|
|(1-x_n)(1-y_n)|
= n -> ∞ as n -> ∞
因此 f 不可能為 uniformly continuous
事實上 f 在每一個不包含 1 的 閉區間 都是一致連續的
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◆ From: 122.127.117.147
※ 編輯: Eliphalet 來自: 122.127.117.147 (03/20 09:30)
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