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. -- 若有發現任何錯誤 還請通知修正 謝謝大家 --

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