课程名称︰分析导论优一
课程性质︰数学系大二必修
课程教师︰王振男
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2014/10/28
考试时限（分钟）：40
试题 :
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.