4. (b) Find the components of 0 under the usual metric. 2. (c) State Monotone Convergence Theorem. Verify the theorem fo
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MMT-004
M.Sc. (MATHEMATICS WITH APPLICATIONS IN COMPUTER SCIENCE) M.Sc. (MACS) Term-End Examination 00395
June, 2014
MMT-004 : REAL ANALYSIS Time : 2 hours
Maximum Marks : 50 Weightage : 70%
Note : Question no. 1 is compulsory. Do any four questions out of question nos. 2 to 7. 1. State, whether the following statements are True or False. Give reasons for your answers. 5x2=10 (a)
If (X, d1) is a discrete metric space and (Y, d2) is any metric space, then any function f : X ---> Y is continuous.
(b)
Continuous image of a Cauchy sequence in a metric space is a Cauchy sequence.
(c)
Discrete metric space has no dense subset.
(d)
Outer measure of a subset of R is always finite.
MMT-004
1
P.T.O.
(e) If f : R2 R is defined by 1+1y1, if(x,y)#(0,0) f(x, y) = 0 , elsewhere then the directional derivative of f along (0, 1) does not exist at (0, 0). 2.
(a) Let d : Rn x Rn —> R be defined by d(x, y) = max 1 x. — y. 1, x, y E R. 1
