课程名称︰分析导论优二
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）︰2015/03/10
考试时限（分钟）：40
试题 :
2
1. (10%) Let Ω be an open set in |R . Assume that (a, b) ∈ Ω and f: Ω → |R
3
is a C function. Prove
4 2π
lim ── ∫ f(a + r cosθ, b + r sinθ) cos(2θ) dθ = f (a, b) - f (a, b)
r→0 2 0 xx yy
πr
2. (10%) Define
α 2 2
╭ |xy| log(x + y ), if (x, y) ≠ (0, 0),
f(x, y) = ╯
╰ 0, if (x, y) ＝ (0, 0),
where α > 1/2. Is f differentiable at (0, 0)?