課程名稱︰分析導論優二
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）︰2015/06/02
考試時限（分鐘）：35
試題 :
1. (10%) Let (x, y) ∈ I = [-1, 1] ×[-1, 1] and
xy
╭ ─────, if (x, y) ≠ (0, 0),
│ 2 2 2
f(x, y) = ╯ (x + y )
│
╰ 0, if (x, y) = (0, 0).
Show that the iterated integrals of f over I are equal, but they are not
equal to the integral of f over I. Does this contradict Fubini's theorem?
γ
2. (10%) Let x f(x) be integrable over (0, ∞) for γ = α and γ = β, where
∞ γ
α < β. Show that for each γ ∈ (α, β), F(γ) = ∫ x f(x) dx exists and
0
is a continuous function of γ.