→ monotones:註：partial符號打不出來，故直接表之140.112.250.148 04/12
課程名稱︰高等微積分
課程性質︰系定必修
課程教師︰王振男 教授
開課系所︰數學系
考試時間︰2005.4.12
試題： Advanced Calculus Midterm 1 (未到者補考版)
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1.(10%) Show that if E is closed in Ｒ and a不屬於E, then
inf ∥x-a∥> 0.
x屬於E
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2.(15%) Suppose that E包含於Ｒ is connected and E包含等於A包含等於E.
Prove that A is connected.
3.A set A is called clopen if and only if it is both open and closed.
(a) (5%) Prove that every Euclidean space has at least two clopen sets.
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(b)(10%) Prove that a proper subset E包含於Ｒ is connected if and only if
it contains exactly two relatively clopen sets.
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(c)(10%) Prove that every nonempty proper subset of Ｒ has a nonempty
boundary.
4.(10%) Prove that each of the following functions has a limit as (x,y) →
(0,0).
(a)
x^3 - y^3
f(x,y) = ──────, (x,y)≠(0,0)
x^2 + y^2
(b) |x|^α‧y^4
f(x,y) = ───────, (x,y)≠(0,0)
x^2 + y^4
where α is ANY positive number.
5.(10%) Prove that f(x,y) = ╭ e^(-1/|x-y|), x≠y
╰ 0, x＝y
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is continuous on Ｒ .
6.(15%) Let B = {(x,y):|x|≦1, |y|≦1} and
f(x,y) = (1/9)x^3 + (1/11)x^2‧(sin y)^3,
g(x,y) = (1/6)y - (1/20)y^2‧(cos x)^5.
Show that there exists only one (a,b) in B such that f(a,b) = g(a,b) = 0.
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7.(15%) Let K be a closed and bounded set in Ｒ (with usual metric, say
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2-norm). Assume that f: K → Ｒ is one-to-one and continupus. Let Ω be
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an open (in Ｒ ) subset of K. Then can f(Ω)包含於partial(f(K))? Prove or
disprove it. Here partial(f(K)) denotes the boundary of f(K).
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