高等微積分的證明題一個題目

Proof

I will show it by contracdition, suppose f(x)≠x for all x in R------(*)

then we have f(x)>x for all x or f(x)<x for all x

Since if there are x', y' with x'<y' s.t f(x')>x' and f(y')<y'

then define g(x)=f(x)-x, g is continuous and g(y')<0, g(x')>0 by 中間值定理

there is x in (x',y') s.t g(x)=f(x)-x=0 contracdits (*)

so we may assume f(x)>x for all x

But we have f(x)>x=f(f(x))>f(x) this is impossible
So there exists a ξ in R such that f(ξ)=ξ

2011-05-22 20:43:57 補充：
以前想到

靠感覺吧

多思考 讓靈感湧現

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小弟不太肯定證明bijective的部分有沒有錯