課程名稱：高等微積分一
課程性質︰數學系必修
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2011/1/11
考試時限（分鐘）：10:20 ~ 13:10
是否需發放獎勵金：是
Choose 5 from the following 9 problems.
(自己選擇五題給助教改，其他題不計分)
編按：(i) φ代表空集合, Ｒ代表實數
(ii) 有小寫字 Bi, fn 的寫法我改成用底線分隔: B_i, f_n
1. Let A and B be two closed subsets of Ｒ^n such that A∩B=φ.
Prove that there exist open sets U and V such that A⊂U, B⊂V,
and U∩V=φ.
2. Let f be a continuous mapping from a metric space X to a metric
space Y.
(a) Prove that f(cl(E))⊂cl(f(E)) for every subset E of X.
(b) Find an example such that f(cl(E))≠cl(f(E)).
(c) Is it true that int(f(E))⊂f(int(E)) for every subset E of X?
3. (a) Prove that if A is connected in a metric space and A⊂B⊂cl(A),
then B is connected.
(b) Show that if a family {B_i} of connected sets satisfies
B_i∩B_j≠φ for all i,j, then ∪B_i is connected.
i
4. Let A be a set in a metric space M.
(a) Show that A is closed and bounded if A is compact.
(b) Show that A is sequentially compact if A is compact.
5. (a) Let f: [0,∞)→Ｒ, f(x) ﹦√x. Use the definition of uniform
continuity to show that f is uniformly continuous on [0,∞).
(b) Show that xcosx is not uniformly continuous on [0,∞).
6. Let (M,d) be a compact metric space and f: M→M satisfy
d(x,y)≦d(f(x),f(y))≦2d(x,y) for all x,y∈M. Show that
d(f(x),f(y))﹦d(x,y) for all x,y∈M and f is a bijection.
7. (a) Let f: [0,1]→Ｒ be defined by f(x)﹦1 if x is irrational and
f(x)﹦0 if x is rational. Show that f is not Riemann integrable
on [0,1].
(b) Let g: [0,1]→Ｒ be a continuous function. Show that g is Riemann
integrable.
8. Determine which of the following sequences of functions converge
pointwise or uniformly as n → ∞.
(a) f_n(x) ﹦ cosx/n on [0,∞).
(b) f_n(x) ﹦ 1/(nx+1) on [0,1].
(c) f_n(x) ﹦ x/(nx+1) on [0,1].
(d) f_n(x) ﹦ nx/((nx)^2+1) on [0,1].
9. Let f: Ｒ→Ｒ be a bounded function. Prove that f is continuous if
and only if the graph {(x,y) | y﹦f(x), x∈Ｒ} of f is a closed subset
of Ｒ^2.
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