Loyola College M.Sc. Mathematics Nov 2006 Real Analysis Question Paper PDF Download


AA 19


         MT 1805 – REAL ANALYSIS



Date & Time : 28-10-2006/1.00-4.00         Dept. No.                                                       Max. : 100 Marks



  1. a)(1) When does the Riemann-Stieltjes integral reduce to Riemann integral. Explain with usual notations.


(2) If a < s < b, f ÎÂ (a) on [a,b] and a (x) = I (x – s), the unit step function, then prove that = f (s).                                                                                                             (5)

b)(1) Let f be a bounded function on [a,b] having finitely many points of discontinuity on [a,b]. Let a be continuous at every point at which f is discontinuous. Prove that f ÎÂ(a).                                                                                                                                                (8)

(2) Suppose f is strictly increasing continuous function that maps an interval [A.B] onto [a,b]. Suppose a is monotonically increasing on [a,b] and f ÎÂ (a) on [a,b]. Define b and g on [A,B] by b (y) = a (f (y)), g (y) = f (f (y)). Then prove that g ÎÂ (b) and .                                                                                                     (7)


(3) Let a be monotonically increasing function on [a,b] and let a¢ Î R on [a,b]. If f is a bounded real function on [a,b] then prove that f ÎÂ (a) on [a,b] Û f a¢ ÎÂ (a) on [a,b].(8)

(4) Let f ÎÂ (a) on [a,b]. For a £ x £ b, define F(x) = , then prove that F is continuous on [a,b]. Also, if f is continuous at some x o Î (a,b) then prove that F is differentiable at x o and F¢ ( x o ) = f (x o ).                                                                           (7)

  1. a) Let : [a,b] ® R m and let x Î (a,b). If the derivatives of exist at x then prove that it is unique.


(2) Suppose that  maps a convex open set E Í Rn into Rm,  is differentiable on E and there exists a constant M such that  M, ” x Î E, then prove that

ú  (b) –  (a)ú £ M ú b – aú , ” a, b Î E.                                                                      (5)


  1. b) (1) Suppose E is an open set in R n ; maps R into R m ; is differentiable at x o Î E,  maps an open set containing    (E) into R k and  is differentiable at f (xo). Then the mapping of E into R k, defined by is differentiable at xo and .                                                                                                  (8)

(2) Suppose  maps an open set EÍ Â n into  m. Let   be differentiable at x Î E, then prove that the partial derivatives (Dj f i) (x) exist and , 1£ j £ m, where {e 1, e  2, e  3, …, e n} and {u 1, u 2, u 3, …, u m} are standard bases of R n and R m.  (7)

(3) If X is a complete metric space and if f is a contraction of X into X, then prove that there exists one and only one x ÎX such that f (x) = x.                                          (15)


III.  a) (1) Prove:  where {f n} converges uniformly to a function f on E and x is a limit point of a metric space E.


(2) Suppose that {f n} is a sequence of functions defined on E and suppose that                  ½f n (x)½£ M n, x ÎE, n = 1,2,… Then prove that converges uniformly on E if converges.                                                                                                                (5)

  1. b) (1) Suppose that K is a compact set and

* {f n} is a sequence of continuous functions on K

** {f n} converges point wise to a continuous function f on K

*** f n (x) ³ f n+1 (x), ” n ÎK, n= 1,2,… then prove that f n ® f  uniformly on K. (7)

(2) State and prove Cauchy criterion for uniform convergence of complex functions defined on some set E.                                                                                                                         (8)


(3) State and prove Stone-Weierstrass theorem.                                                            (15)

IV a) (1)Show that  converges if and only if n >0.


(2) Prove that G  = .                                                                                         (5)

b)(1) Derive the relation between Beta and Gamma functions.                                       (7)

(2) State and prove Stirling’s formula.                                                                          (8)


3) If f is a positive function on (0,¥) such that f (x+1) = x f (x);  f (1) =1 and log f is convex then prove that f (x) = G (x).                                                                                               (8)

(4) If x >0 and y >0  then                                        (7)

  1. a) (1)If f (x) has m continuous derivatives and no point occurs in the sequence x 0, x 1, ..,x n more than (m+1) times then prove that there exists exactly one polynomial Pn (x) of degree £ n which  agrees with f (x) at x 0, x 1, …, x n.


2) Show that the error estimation for sine or cosine function f in linear interpolation is given by the formula ½f(x)-P(x)½£ .                                                                    (5)

b)(1) Let x0, x1, …, xn be n+1 distinct points in the domain of a function f and let P be the interpolation polynomial of degree £ n, that agrees with f at these points. Choose a point x in the domain of f and let [a,b] be any closed interval containing the points x 0, x 1, …, x n  and x. If f has a derivative of order n+1 in the interval [a,b], then prove that there is at least one point c in the open interval (a,b) such that  where A (x) = (x – x0) (x – x1)…(x – x n).            (7)

(2) Let P n+1 (x)= x n+1 +Q(x) where Q is a polynomial of degree £ n and let maximum of ½P n+1 (x)½, -1 £ x £ 1.  Then prove that we get the inequality . Moreover , prove that   if and only if , where T n+1 is the Chebyshev polynomial of degree n+1.                                          (8)


3) Let f be a continuous function on [a,b] and assume that T is a polynomial of degree £ n that best approximates f on [a,b] relative to the maximum norm. Let R(x) = f (x) –T(x) denote the error in the approximation and let D = . Then prove that

(i) If D= 0 the function R is identically zero on [a,b].

(ii) If D >0, the function R has at least (n+1) changes of sign on [a,b].              (15).



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