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Question Number 414 by 123456 last updated on 25/Jan/15
f(x,y)= { ((((x−1)(y−1)(xy−1)+(x^2 −1)(y^2 −1))/(x^2 −xy)),((x,y)≠(1,1))),(0,((x,y)=(1,1))) :} is f(x,y) continuos at (x,y)=(1,1)?
$${f}\left({x},{y}\right)=\begin{cases}{\frac{\left({x}−\mathrm{1}\right)\left({y}−\mathrm{1}\right)\left({xy}−\mathrm{1}\right)+\left({x}^{\mathrm{2}} −\mathrm{1}\right)\left({y}^{\mathrm{2}} −\mathrm{1}\right)}{{x}^{\mathrm{2}} −{xy}}}&{\left({x},{y}\right)\neq\left(\mathrm{1},\mathrm{1}\right)}\\{\mathrm{0}}&{\left({x},{y}\right)=\left(\mathrm{1},\mathrm{1}\right)}\end{cases} \\ $$$$\mathrm{is}\:{f}\left({x},{y}\right)\:\mathrm{continuos}\:\mathrm{at}\:\left({x},{y}\right)=\left(\mathrm{1},\mathrm{1}\right)? \\ $$
Answered by prakash jain last updated on 31/Dec/14
The function is not continuous since limit does not exist on line x=y.
$$\mathrm{The}\:\mathrm{function}\:\mathrm{is}\:\mathrm{not}\:\mathrm{continuous} \\ $$$$\mathrm{since}\:\mathrm{limit}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist}\:\mathrm{on} \\ $$$$\mathrm{line}\:{x}={y}. \\ $$

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