Anna University Chennai 2007 M.Sc Computer Science Real Analyasis - Question Paper

Register Number :
Name of the Candidate :
7 seven 0 3
M.Sc. DEGREE EXAMINATION, 2007
( REAL ANALYSIS )
( 1st YEAR )
( PAPER - two )

520. REAL ANALYSIS
( New Regulations )
May ] [ Time : three Hours
Maximum : 100 Marks

part Œ A (8 × five = 40)

ans any 8 ques.. All ques. carry equal marks.
1. describe derivative of a real valued function at a point c Î(a, b). If f is differentiable at c Î (a, b), prove that f is continuous at c Î (a, b).
2. State and prove generalized Mean ΠValue theorem.

3. If P ¢ is finer than P, prove that U(p ¢ , f, a ) £ U(P, f, a )
4. If f is continuous on [a,b] and if a is of bounded variation on [a, b], prove that f Î R( a ) on [a, b].
5. State and prove 2nd fundamental theorem of integral calculus.
6. State and prove Weierstrass M-lest.
7. Show that :
8. Prove that the interval (a, ¥ ) is measurable.
9. If f is a non-negative integral function over a set E, prove that provided Î > 0 there exists d > 0 such that for every set A Ì E with m(A) < d
we have
10. If f is a bounded measurable function on [a, b]
and F(n) = f(t)dt + F(a), prove that
F ¢ (n) = f(n) a.e.
part Œ B (3 × 20 = 60)
ans any 3 ques.. every ques. carries equal marks.
11. (a) State and prove Taylor™s formula. (10)
(b) Let f be of bounded variation on [a, b] and let V(x) = V f (a, x), V(a) = 0 for x Î (a, b). Prove that V is continuous if a is continuous. (10)
12. If f Î R( a ) on [a, b] and if a has a continuos derivative a¢ on [a,b], prove that
exists
and
(20)
13. (a) State and prove the change of variable in Riemann integral. (8)
(b) State and prove cauchy condition for uniform convergence of sequence of functions. (12)


14. (a) Prove that the outer measure of an interval is its length. (15)
(b) If m x (E) = 0, Prove that E is measurable. (5)
15. (a) State and prove bounded convergence theorem. (10)
(b) State and prove Lebesgue convergence theroem. (10)

Earning:   Approval pending.