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C 20645 (Pages : 3) Name.........................................
Reg. No.....................................
SIXTH SEMESTER U.G. DEGREE EXAMINATION, MARCH 2022
(CBCSS–UG)
Mathematics
MTS 6B 10—REAL ANALYSIS
(2019 Admissions)
Time : Two Hours and a Half Maximum : 80 Marks
Section A
Answer at least ten questions.
Each question carries 3 marks.
All questions can be attended.
Overall Ceiling 30.
1. Define continuity of a function. Show that the constant function
f x b
is continuous on
\
.
2. State Boundedness theorem. Is boundedness of the interval, a necessary condition in the theorem ?
Justify with an example.
3. If
: A IR
f
o
is uniformly continuous on A
\
and (xn) is a Cauchy sequence in A. Then show
that f (xn) is a Caychy sequence in
\
.
4. Define Riemann sum of a function
>
@
: ,f a b o
\
.
5. Suppose f and g are in
>
@
,
a b
\
then show that f + g is in
>
@
,
a b
\
.
6. State squeeze theorem for Riemann integrable functions.
7. If f belong to
>
@
,
a b
\
, then show that its absolute value |f | is in
>
@
,
a b
\
.
8. Define pointwise convergence of a sequence (fn) of functions.
9. Discuss the uniform convergence of
on ( 1,1].
n
n
f x x 10. If
2
2
nx
n
h x nxe for
>
@
0,1 ,x n
`
and
0
h x
for all
>
@
0,1
x, then show that :
1 1
0 0
lim .
n
h x dx h x dx
z³³11. State Cauchy criteria for uniform convergence series of functions.
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