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Total No. of Pages: 1

Register Number: 6174

Name of the Candidate:

M.Sc. DEGREE EXAMINATION – 2013

(MATHEMATICS)

(FIRST YEAR)

(PAPER-II)

120. REAL ANALYSIS

May) Maximum: 100 Marks (Time: 3 Hours

SECTION-A

Answer any EIGHT Questions (8×5=40)

All questions carry equal marks

1.  / State and prove the Rolle’s theorem.
2.  / If f is monotomic on [a, b] prove that the set of discontinuities of f is countable.
3.  / Assume that α­[a, b]. If fÎR(α) on [a, b], prove that |f| ÎR(α) on [a, b] and we have
4.  / State and prove the first mean value theorem for Riemann-Stieltjes integrals.
5.  / Assuming that fnàf uniformly on S. If each fn is continuous at a point c of s, prove that the limit functions f is also continuous at C.
6.  / State and prove the Weierstrass M- test.
7.  / State and prove the Bounded Convergence theorem.
8.  / Let c be a constant and f and g two measurable real-valued function defined on the same domain. Prove that the functions f+c, cf, f+g, g–f and fg are also measurable.
9.  / If an is real and positive for every value of n, prove that the product π(1+an) converges or diverges according as the series ∑an converges and diverges.
10.  / Prove that the product converges absolutely for every value of x, real or complex.

SECTION-B

Answer any THREE Questions (3×20=60)

All questions carry equal marks

11.  / a)  State and prove the chain rule for differentiation
b)  Let f be of bounded variation on [a, b]. Let V be defined on [a, b] as V(x)=Vf(a, x) if a<x£b, V(a)=0. Prove that (i) V is an increasing function on [a, b] (ii) V-f is an increasing function on [a, b].
12.  / a)  If f ÎR(α) on [a, b] prove that αÎR(f) on [a, b] and that
b)  Assume that α­on [a, b] . Prove that the following conditions are equivalent.
(i)  fÎR(α)on [a, b]
(ii)  f satisfies Riemann’s condition with respect to α on [a, b]
(iii)  I(f, α)=(f, α)
13.  / a)  State and prove the Bernstein theorem.
b)  Let α be of bounded variation on [a, b]. Assume that each term of the sequence {fn} is a real-valued function such that fnÎR(α) for each n=1, 2, ...... Assume that fnàf uniformly on [a, b] and define gn(x) = if xÎ[a, b], n=1, 2,.... then prove that (i) fÎR(α) on [a, b] (ii) gnàg uniformly on [a, b] where g(x) =
14.  / a)  Let E be a set of finite outer measure and I a collection of internals that cover E in the sense of vitali. Prove that given Î>0, there is a finite disjoint collection {I1, ...... ,IN} of intervals in I such that m*[E~]<t.
b)  Define absolute continuity of a function f on [a, b]. If f is absolutely continuous on [a, b] and f ¢(x)=0 a.e prove that f is constant on [a, b].
15.  / a)  Show that
b)  Prove that cot h x =

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