Elementary Mathematical Analysis — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

3. Each part carries equal marks. 1. (a) Prove that lim f(x) = L if and only if for all x →0 sequences (x) in Dit) - {o›, such that i*"› (b) Using Sequential Criterion of Limits prove that lim sin- does not exist. (c) Define continuity of function at a point. Prove x x # 0 that the Signum function sgn (x) = X is (a) Prove that any continuous function F: [a, b] → R must satisfy the Intermediate Value Property on [a, b], that is for all y between f(a) and f(b) 3 c € [a, b] such that f(c) = y. (b) Define uniform continuity of a function f on an interval I. Prove that the function f(x) =- is not uniformly continuous on (0,1). (c) For any partitions P and Q of [a, b], prove that (a) Use sequential criterion to show that the function f(x) = x2 - 2x is integrable on [2,5]. (b) Consider the characteristic function of a single-point set, say f = (s) defined by (x) = 1, if x =5 l 10, Otherwise) Prove that f is integrable on 12,91 and find fit. (c) State Fundamental Theorem of Calculus. Use it to evaluate S'° -10°gn (x) dx. 4. (a) Prove that |f] is integrable on [a,b] for a function f integrable on [a,b]. Also prove that S° where where Mis an upper bound for |f| on [a,b]. (b) For the function f(x) = x2 + 3x, find S(f, P) and 5((P), where P= 055 2 is a partition of (c) Find pointwise limit of the following sequence of functions: 5. (a) Check whether the following sequence of functions are uniformly convergent on the specified domains. (b) Show that the sequence of functions is uniformly convergent on [O, 0), where f, (x) = - on [0, ∞0) . (c) Suppose <f,> converges pointwise to fon a subset S of R. It each f, has a limit at xo for some o e S. Does f have a limit at x? Justify your claim. 6. (a) State Weierstrass M-Test for uniform convergence for a series of functions. Use it to check the uniform convergence of the following : (b) Find radius of convergence and interval of convergence of the following power series: (c) Show by integrating that the series for 1+x2