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）：2014/11/18
考試時限（分鐘）：40
試題 :
1. (10%) A function f of two real variables is defined for each point (x, y) in
the unit square [0, 1] ×[0, 1] as follows:
╭ 1, x ∈ |Q,
f(x, y) = ╯ c
╰ 2y, x ∈ |Q .
_1 1
(a) Compute ∫ f(x, y) dx and ∫ f(x, y) dx in terms of y.
0 ￣0
1 t
(b) Show that ∫ f(x, y) dy exists for each fixed x and compute ∫ f(x,y) dy
0 0
in terms of x and t for x ∈ [0, 1] and t ∈ [0, 1].
1 1
(c) Let F(x) = ∫ f(x, y) dy. Show that ∫ F(x) dx exists and find its value.
0 0
2
2. (10%) Assume that f is a bounded real function on [a, b]. Does f ∈ R on
3
[a, b] imply f ∈ R on [a, b]? Does f ∈ R on [a, b] imply f ∈ R on [a, b]?