Re: [分析] 連續性的問題

[Math ()]

※ 引述《Rudin5566 (魯丁.嗚嗚溜溜)》之銘言：
: If 1.f is continuous on [a b]
:    2.F(x)=supf([a x])
: Prove that F is continuous on [a b]

Since f is continuous on compact set [a,b], f is uniformly continuous on [a,b].

Given ε>0 there is a δ>0 such that

        |f(x)-f(y)|<ε whenever 0<|x-y|<δ.

WLOG assume y>x.

For 0<|y-x|<δ, sup f([a,y]) = sup{ sup f([a,x]) , sup f([x,y]) }.

If sup f([a,y]) = sup f([a,x]), then F(y)-F(x)=0<ε.

If sup f([a,y]) = sup f([x,y]). Since f is continuous on [x,y], then

there exists y' in [x,y] such that sup f([x,y]) = f(y').

        ε > f(y') - f(x) ≧ f(y') - sup f([a,x]) = F(y) - F(x) ≧ 0 > -ε

Thus |F(y)-F(x)|<ε whenever 0<|x-y|<δ □

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※ 編輯: Math            來自: 118.168.166.97       (11/16 04:08)
推 ic6413      :Math 還是比 Rudin 厲害XD