作者mercedesff (mercedesff)
看板trans_math
标题[单变] 几题，出在"极限与连续"章节中
时间Sat Dec 4 21:58:57 2010
以下想破头，仍不得其解，请大大帮忙(跪)，谢谢!
╭ 1, if x=1;
1. Let f(x) = ┤
╰ 2, if x=2.
Is f a continuous function on its domain? Justify your answer.
2. Suppose that f:[a,b]→R is a continuous function.
(a) Let a ≦ x_1 ＜ x_2 ≦ b . Prove that there is a point z 属於 [a,b]
such that f(x_1) + f(x_2)
f(z) = 一一一一一一一一一一一 .
2
(b) Let a ≦ x_1 ＜ x_2 ＜ x_3 ≦ b . Prove that there is a point
z 属於 [a,b] such that f(x_1) + f(x_2) + f(x_3)
f(z) = 一一一一一一一一一一一一一一一 .
3
3. Let f:(-∞,∞) → R be a continuous function such that there are
two real numbers a and b, such that f(a) ≦ f(x) ≦f(b) for
all -∞ ＜ x ＜ ∞ . Is the equation f(x) - x = 0 solvable ?
～～～～～
Justify your answer.
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※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 140.113.90.123
1F：→ kane950544:2.Let F(x)=f(x)-0.5[f(x1)+f(x2)] 114.44.186.94 12/06 21:44
2F：→ kane950544:设f(x1)>f(x2),再用中间值定理 114.44.186.94 12/06 21:46
3F：→ kane950544:b部分就取max和min之後用中间值定理吧 114.44.186.94 12/06 21:47
4F：→ kane950544:Let M=Max(f1,f2,f3),m=min(f1,f2,f3) 114.44.186.94 12/06 21:48
5F：→ kane950544:Is the equation f(x)-x=0是在问什麽? 114.44.186.94 12/06 21:49
对不起，题目少打一个字，已改於上面了
※ 编辑: mercedesff 来自: 140.113.90.123 (12/08 18:30)
6F：→ kane950544:那就设F(x)=f(x)-x 利用f(x)的有界性 114.24.172.124 12/09 23:40
7F：→ kane950544:F(x)→-∞ as x→∞ 114.24.172.124 12/09 23:42
8F：→ kane950544:F(x)→∞ as x→-∞ 114.24.172.124 12/09 23:42
9F：→ kane950544:由中值定理就可以了 114.24.172.124 12/09 23:43
10F：→ kane950544:第一题定义域只有两个点 我也看不懂 114.24.172.124 12/09 23:44