作者Dirichlet ( )
看板Math
标题Re: [分析] 初微(44)
时间Tue Aug 2 15:49:31 2005
※ 引述《TaiwanBank (澳仔金控台湾分行)》之铭言：
: Suppose that f: (-5,5) → R is a continuous function. Which
: of the following statements must be true and which could be
: false? (Give reasons for your answers)
: (a) The set { f(x) : 0 < x < 1 } is open.
: (b) The set { f(x) : 0 < x < 1 } is bounded.
: (c) f is uniformly continuous on the interval (-1,1).
(a) False, suppose f(x)=0 => the set { f(x) : 0 < x < 1 } = {0} is closed
(b) 连续函数保持紧致性 => the set { f(x) : 0 ≦ x ≦ 1 } is compact
这导致 { f(x) : 0 < x < 1 } 有界
(c) 连续函数 f 在紧致集 [1,-1] 上是均匀连续
由定义可推得在 (-1,1) 上亦均匀连续
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