[單變] 幾題，出在"極限與連續"章節中

[mercedesff (mercedesff)]

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.

--
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◆ From: 140.113.90.123
→ kane950544:2.Let F(x)=f(x)-0.5[f(x1)+f(x2)]
→ kane950544:設f(x1)>f(x2),再用中間值定理
→ kane950544:b部分就取max和min之後用中間值定理吧
→ kane950544:Let M=Max(f1,f2,f3),m=min(f1,f2,f3)
→ kane950544:Is the equation f(x)-x=0是在問什麼?
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)
→ kane950544:那就設F(x)=f(x)-x 利用f(x)的有界性
→ kane950544:F(x)→-∞ as x→∞
→ kane950544:F(x)→∞ as x→-∞
→ kane950544:由中值定理就可以了
→ kane950544:第一題定義域只有兩個點 我也看不懂