Kannur University 2011-1st Sem M.Sc Mathematics ( Syllabus) MODEL MAT1C03: Real Analysis - - Question Paper

PART-A
1. Verify whether ??????? ?????
??????? ??? ?? , is a metric on ?G
2. Let ? ??? ? be continuous mappings of a metric space ? into a metric space ?and ? a
dense subset of ?. If ????????? ??? ??? ? ??? prove that????????? ??? ??? ? ??G
3. Show that L'Hospital's rule does not hold for vector-valued functions.
4. If ????????? calculate ????????????? for all real ?. Show that ??????? does not exists.
5. Examine whether the function provided by ???????????
?????? ??????????, is of
bounded variation on [0, 1].
PART-B
6. Show that for a subset E of ??? subsequent statements are equivalent
i. E is closed and bounded
ii. E is compact
iii. Every infinite subset of E has a limit point in E.
.
7. a) Let ? ??? ? be metric spaces .Show that a function ????? is continuous if and only
if ?????? is closed in ? for every closed set ? in ?.
b) If ??????g?? are the coordinates of a point ??? , show that the functions ?? described
by ????????? ? ??? are continuous.
8. a) Let ? be described on ?????and is differentiable at ? ?????? .Show that ? is
continuous at ?.
b) State and prove the generalized mean value theorem.
9. a) Show that ? ????? on ????? if and only if for every r ??? there exists a partition P
of ????? such that ???????????????????.
b) If ? is monotonic on????? and ? is continuous on ????? show that ? ? ????
10. a) Prove the Fundamental theorem of calculus.
b) Let ? ? ? on ????? and ????? ???????? ??? ?????
? .Show that ? is
differentiable at a point ?? ?????? whenever ? is continuous at ??.
11.a) Let ? be a monotonic function on [a,b].Show that ? is of bounded variation on [a,b]
b) Let ?????????? be a path with components ?????????gG??? Show that ? is
rectifiable if and only if every component ?? is of bounded variation on ?????G
PART-C
UNIT I
12. a) Let P be a nonempty perfect set in??G Show that P is uncountable.
b) Show that a subset ? of the real line ? is connected if and only if ??? ?? ???
????? ?????? ???? ? ??.
13. a) Show that a continuous function ? from a compact metric space? into a metric space
? .is uniformly continuous on ?G
b) Let ? be monotonic on?????G Show that the set of points of ????? at which ? is
discontinuous is at most countable.
UNIT II
14. a) State and prove the Taylor's Theorem
b) Let ? be a continuous mapping of ????? into ?? which is differentiable in ?????G
Show that there exists ??????? such that ????????????????????????
15. a) If ??? ?????, show that ??? ????? and ????????? ????? ????
b) Let ? be of bounded variation on ??????? is monotonically increasing and ???? on
?????G Show that ????? ??????
?
?
? ???????.
UNIT III
16. a) Let ????????????? ? ? ???? for a few monotonically increasing function
? ?? ?????G Show that ??? ????? and ? ?????
? ?? ?????
? ??.
b) Let ? be continuous on????? .Show that ? is of bounded variation on ????? if and
only if ? can be expressed as the difference of 2 increasing continuous
functions.
17. a) Let ? and ? be differentiable functions on ????? such that
???? ?? ??? ???? ??G Show that
? ?????????????????????????? ??????????G
?
?
?
?
b) If ? is of bounded variation on ?????, show that ? is bounded on ?????.
c) Let ? be a rectifiable path on ????? of arc length??????? and ? ??????.Show
that ???????????????? ???????G

Earning:   Approval pending.