课程名称︰实分析二
课程性质︰数学研究所必选修
课程教师︰刘丰哲
开课学院：理学院
开课系所︰数学系、数学研究所、应用数学科学研究所
考试日期（年月日）︰104.4.22
考试时限（分钟）：100min
试题 :
1.(10%) Let f be a BV function on [a,b]. Show that f is AC if and only if
b b
V (f)＝∫|f'|dλ.
a a
2.(15%) Let f be a continuous function defined on a finite open interval (a,b)
such that f is AC on every closed interval in (a,b) and f' is integrable on
(a,b). Show that f can be extended to be an AC function on [a,b].
n
3.(10%) Suppose K⊂Ω⊂R, where K is compact and Ω is open. Show that there
∞
is u∈C (Ω) such that 0≦u≦1 and u＝1 on K.
c
n
4.Let f be a locally integrable function defined on an open set Ω in R.
n
(i)(10%) Show that f＝0 a.e. on Ω if ∫fdλ ＝0 for all compact set K in Ω.
n K ∞
(ii)(10%) show that if ∫ fφdλ ＝0 for all φ∈C (Ω), then f＝0 a.e. on Ω.
Ω c
5.Suppose that t is a continuously differentiable map from an open set Ω in
n n
R into R.
n
(i)(10%) Show that λ (tA)＝0 if A is a null set in Ω.
(ii)(12%) Let D={x∈Ω：J(t;x)=0} , where J(t;x) is the Jacobian determinant
n
of t at x. Show that λ (tD)=0.
6.(8%) Let f be the Cantor's ternary function on [0,1] and P the Cantor's tern-
1
ary set in [0,1]. Find ∫fdf and ∫fdμ , where μ is the Lebesgue-Stieltjes
0 P f f
measure on [0,1] generated by f.
7.(15%) Show that
t 　2x ∞ j t 2j+1
∫ 一一 dx＝2Σ(-1) ∫ x dx
0 1+x^2 j＝0 0
for 0＜t＜1 ; then show that
1 2x ∞ j 1
∫ 一一 dx＝Σ(-1) 一一
0 1+x^2 j＝0 j+1
∞ j 1
and evaluate Σ(-1) 一一.
j＝0 j+1