推 t0444564 : 已收錄 10/22 04:27
※ 編輯: xavier13540 (140.112.249.76), 10/24/2014 00:42:38
→ xavier13540 : 2.的"n>2"改成"n>1"(雖然沒差拉XD) 10/24 00:43
推 t0444564 : 已針對修改 10/24 01:03
課程名稱︰分析導論優一
課程性質︰數學系大二必修
課程教師︰王振男
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014/10/21
考試時限（分鐘）：190
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. (a) (10%) Prove that Cantor's Intersection Theorem holds for nested compact
sets in an arbitrary metric space, namely, if K , K , ... is a nested
1 2
sequence of nonempty compact sets in a metric space S, then
∞
∩ K ≠ψ
j=1 j
(b) (10%) In (a), can we replace "compact sets" by "closed and bounded
sets"?
n n 2
2. Let S = {x = (x , x , ..., x ) ∈ |R (n > 1) : Σ x = 1}. Assume that f
1 2 n k=1 k
is continuous function on S.
(a) (10%) If f(x)≠0 for all x∈S and f(x ) > 0 for some x ∈S. Show that
0 0
there exists a constant c > 0 such that f(x) ≧ c for all x∈S.
(b) (10%) Show that there exists a pair of diametrically opposite points on
S at which f assumes the same value, i.e., for some x∈S, f(x) = f(-x).
3. (a) (10%) Let f: S → T be a continuous function. Assume that S is compact.
Show that for any Cauchy sequence {x } in S, {f(x )} is a Cauchy
n n
sequence in T.
(b) (10%) Does the result in (a) remain true if we remove the compactness
assumption on S?
4. (20%) Let f be a real-valued function defined on [a, b]. The graph of f on
[a, b] is given by
2
G(f) = {(x, f(x))∈|R : x∈[a, b]}
Show that f is continuous on [a, b] iff G(f) is closed and connected in
2
|R .
5. (a) (10%) Let (M, d) be a metric space, define
d(x, y)
d'(x, y) =──────
1 + d(x, y)
Prove that d' is also a metric for M.
(b) (10%) Is any open set in (M, d) open in (M, d') and vice versa?
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第01話 似乎在課堂上聽過的樣子 第02話 那真是太令人絕望了
第03話 已經沒什麼好期望了 第04話 被當、21都是存在的
第05話 怎麼可能會all pass 第06話 這考卷絕對有問題啊
第07話 你能面對真正的分數嗎 第08話 我，真是個笨蛋
第09話 這樣成績，教授絕不會讓我過的 第10話 再也不依靠考古題
第11話 最後留下的補考 第12話 我最愛的學分
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※ 編輯: xavier13540 (140.112.249.76), 10/21/2014 21:45:16
※ 編輯: xavier13540 (140.112.249.76), 10/21/2014 21:57:31