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MTH631 Real Analysis II Assignment 2 Solution & Discussion Fall 2019

Assignment # 02                         MTH631 (Fall 2019)

Maximum Marks:  20                 Due Date: 20 -01-2020


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Q1. Use Bolzano-Weierstrass theorem to show that if{S_1},{S_2},...,{S_m},...is an infinite sequence of nonempty compact sets{S_1} \supset {S_2} \supset ... \supset {S_m} \supset ...,and then\bigcap\limits_{m = 1}^\infty {{S_m}} is nonempty. Show that the conclusion does not follow if the sets are assumed to be closed rather than compact.


Q2. Letf(x,y) = ({x^4}y - ?2x{?^3}{y^2} + 3{x^2}{y^3} + {y^5})/{({x^2} + {y^2})^2}when (x,y)≠(0,0) and f(x,y)=0 when (x,y)=(0,0). Determine {\lim }\limits_{(x,y) \to (0,0)} f(x,y)Is function continuous at the origin?

MTH631 Assignment No 02 Solution Fall 2019

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