作者mercedesff (mercedesff)
看板trans_math
標題[單變] 幾題，出在"極限與連續"章節中
時間Sat Dec 4 21:58:57 2010
以下想破頭，仍不得其解，請大大幫忙(跪)，謝謝!
╭ 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 屬於 [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 屬於 [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.
--
※ 發信站: 批踢踢實業坊(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:設f(x1)>f(x2),再用中間值定理 114.44.186.94 12/06 21:46
3F：→ kane950544:b部分就取max和min之後用中間值定理吧 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是在問什麼? 114.44.186.94 12/06 21:49
對不起，題目少打一個字，已改於上面了
※ 編輯: mercedesff 來自: 140.113.90.123 (12/08 18:30)
6F：→ kane950544:那就設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:第一題定義域只有兩個點 我也看不懂 114.24.172.124 12/09 23:44