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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 differentiationb) 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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