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年04月10日 14:00-17:00
試題： Advanced Calculus First Midterm (A)
1.(15%) Suppose that {E_α} α∈Α is a collection of connected sets in
n
a Euclidean space Ｒ such that ∩ E_α≠ψ. Prove that
α∈Α
E = ∪ E_α
α∈Α
is connected.
2.(20%) Compute the iterated limits at (0,0) of each of the following func-
tions. Determine which of these functions has a limit as (x,y) → (0,0)
n
in Ｒ , and prove that the limit exists.
(a)
sinx‧siny
f(x,y) = ───────.
x^2 + y^2
(b)
x^2 + y^4
f(x,y) = ───────.
x^2 + 2y^4
(c)
x - y
f(x,y) = ───────, α< (1/2)
(x^2 + y^2)^α
n m
3.(15%) Let B be closed in Ｒ and f: B → Ｒ . Prove that the following are
equivalent:
(a) f is continuous on B.
-1 n m
(b) f (E) is closed in Ｒ for every closed subset E of Ｒ .
n
4.(20%) Let n,m∈Ｎ, E包含於Ｒ , and suppose that D is dense in E. If
m
f:D → Ｒ is uniformly continuous on D, prove that f has a continuous
m
extension to E; ie., prove that there is a continuous function g: E → Ｒ
such that g(x) = f(x) for all x∈D.
5.Let X be a compact metric space with metric ρ. Let f:X → X be a function
satisfying
ρ(f(x), f(y))≦ (1/2)ρ(x, y), for all x,y屬於X
Can there exist x_0∈X such that f(x_0) = x_0? (10%) Can the space X has
completeness? (5%) Prove or disprove all your answers.
n n 1
6.(10%) Suppose that f: V → Ｒ and V is open in Ｒ . Let f∈C (V) and
Df(x) is invertible for all x∈V. Then is f one-to-one on V when n > 1?
Please answer the same question when n = 1.
7.(5%) Yes and no questions.
(i) In any metric space, compactness implies closedness and boundedness.
(ii) In any metric space, closedness and boundedness implies compactness.
(iii) In Euclidean space with usual metric, sequential compactness, com-
pactness, closedness and boundedness, are equivalent.
n n
(iv) Let ρ be a metric in Ｒ . Then Ｒ is always unbounded in the metric
n
space (Ｒ , ρ).
(v) Existence of all first order partial derivatives implies continuity.
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