作者axis0801 (型男要考试)
看板trans_math
标题Re: [微分] 95政大应数3
时间Sun Nov 19 02:45:23 2006
: ※ 引述《pobm (待从头收拾旧河山)》之铭言：
: : If f(x) is continuous on [0,1]
and satisfies 0≦f(x)≦1, then f has a fixed point
^^^^^^^^^^^^^^^^^^^^^^^^ 题目有瑕疵..必须加上这个条件！
: : show that there exists c in [0,1], such that f(c)=c
既然题目要求满足f(c)=c, 已知 Df on [0,1], 则势必 0≦ Rf ≦1
＜pf＞：
let g(x)＝f(x)－x is continuous on [0,1]
so we have g(0)＝f(0)－0≧0, g(1)＝f(1)－1≦0 for 0≦f(x)≦1
∴ g(0)g(1)≦0
By I.V.T Then there is c in [0,1] s.t g(c)＝0
=> f(c)－c＝0 ∴ f(c)＝c
Q.E.D
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※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 61.62.121.187
1F：推 GayerDior:还有两个case (0，0)点和(1，1)点 61.229.153.246 11/19 08:25
2F：→ GayerDior:3个case综合才满足c in [0,1] 61.229.153.246 11/19 08:26
3F：→ GayerDior:不然你酱算只满足c in (0,1) 61.229.153.246 11/19 08:26
4F：推 axis0801:注意g(0)g(1)≦0 就已包含(0,0)和(1,1) 61.62.121.187 11/19 08:49
5F：→ axis0801:因此c=0满足g(0)=0, 或c=1满足g(1)=0 61.62.121.187 11/19 08:53
6F：→ axis0801:既然有g(0)≧0, g(1)≦0的条件了,当然也 61.62.121.187 11/19 08:57
7F：→ axis0801:就暗示了f(0)=0 和 f(1)=1, 所以我才能有 61.62.121.187 11/19 09:05
8F：→ axis0801:足够条件说必存在 c in [0,1] s.t g(c)=0 61.62.121.187 11/19 09:07
9F：推 GayerDior:嗯嗯 没注意看 ;p 61.229.159.204 11/19 11:09