Padmashree Dr DY Patil Vidyapeeth 2007 M.Sc Mathematics Mathamtics - university paper

M.Sc. DEGREE EXAMINATION, 2007

( MATHEMATICS )

( 1st YEAR )

( PAPER - II )

120. REAL ANALYSIS

( Revised Regulations )

May ] [ Time : three Hours

Maximum : 100 Marks

PART ? A (8 × five = 40)

ans any 8 ques..

All ques. carry equal marks.

1. State and prove generalized Mean ? Value

theorem.

2. If f and g are of functions of bounded

variation on [a, b], prove that f + g is of

bounded variation.

3. If P¢ is a refinement of P, Prove that

U (P¢, f, a ) £ U(p, f, a)

4. If fÎÂ(a), prove that for every t > 0

there exists a partition P such that

U (P, f, a) ? L (P, f, a ) < t.

5. provide an example to show that a sequence of

continuous functions need not converge to a

continuous function.

6. State and prove Weierstrass M-test.

7. Prove that [0, 1] is uncountable.

8. If f is of bounded variation [a, b], prove that

f (b) ? f (a) = Pab ? Nab.

9. If (1 + an) converges absolutely,

prove that it converges.

10. obtain the value of (1 ? n?2).


PART ? B (3 × 20 = 60)

ans any 3 ques..

All ques. carry equal marks.

11. (a) State and prove chain rule for differentiation.

(10)

(b) Prove that f is of bounded variation on

[a, b] iff f can be expressed as the difference

of 2 increasing functions. (10)

12. Let a be of bounded variation on [a, b].

Let V(n) be the total variation of a on

[a, x] and let V(a) = 0. If f ÎÂ(a),

prove that f ÎÂ (V) ? (20)

13. (a) If fn Î Â(a) for every n, if fn ® f uniformly

on [a, b] and if gn(n) = fn (t) da(t),

prove that f ÎÂ(a) and gn ® g uniformly

where g(n) = f(t) da (t) (10)

(b) State and prove Bernstein theorem. (10)

14. (a) State and prove Littlewood?s 3rd principle.

(10)

(b) Prove that every absolutely continuous

function is the indefinite integral of its

derivative. (10)

15. (a) If an ³ 0 n, prove that p (1?an)

converges iff the series å an converges.

(10)
(b) State and prove Cauchy condition for

infinite product. (10)

Earning:   Approval pending.