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）︰2014/10/28
考試時限（分鐘）：40
試題 :
1. (10%) Consider the function f defined on |R by
╭ 0 if x irrational,
│
f(x) = ╯
│ 1 m
╰ ─ if x = ─,
n n
where m and n do not have common divisors and n = 1 if x = 0. Prove that f is
continuous at every irrational point, and that f has a jump discontinuity at
every rational point.
2. (10%) Let K and C be two disjoint sets in a metric space X. Assume that K is
compact and C is closed. Prove that there exists a δ > 0 such that d(x, y) >
δ for x ∈ K and y ∈ C.