LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOV 2006
MT 1805 – REAL ANALYSIS
Date & Time : 28102006/1.004.00 Dept. No. Max. : 100 Marks
 a)(1) When does the RiemannStieltjes integral reduce to Riemann integral. Explain with usual notations.
OR
(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)
OR
(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)
 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.
OR
(2) Suppose that maps a convex open set E Í R^{n} into R^{m}, 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)
 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 (x_{o}). Then the mapping of E into R ^{k}, defined by is differentiable at x_{o} and . (8)
(2) Suppose maps an open set EÍ Â ^{n} into Â ^{m}. Let be differentiable at x Î E, then prove that the partial derivatives (D_{j }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.
OR
(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)
 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)
OR
(3) State and prove StoneWeierstrass theorem. (15)
IV a) (1)Show that converges if and only if n >0.
OR
(2) Prove that G = . (5)
b)(1) Derive the relation between Beta and Gamma functions. (7)
(2) State and prove Stirling’s formula. (8)
OR
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)
 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 P_{n} (x) of degree £ n which agrees with f (x) at x _{0}, x _{1}, …, x _{n}.
OR
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 x_{0}, x_{1}, …, x_{n} 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 – x_{0}) (x – x_{1})…(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)
OR
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).