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I need to find the value of $a$ for which the function $f(x,y)= \frac{x^2-y^2}{x^2+y^2}$ if $(x,y) \neq (0,0) $ and $f(x,y)=a$ when $(x,y)=(0,0)$ when continuous along the path $y=b\sqrt{x}$ where $b\neq 0$ and then $f$ is continuous.

The answer I obtained for the first part is $a=-1.$ Is this correct? What about the second part ?

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    $\begingroup$ Your "second part" is not clear. Do you mean making the function $f$ continuous in both variables at all points? $\endgroup$ – Rory Daulton Apr 18 '15 at 15:16
  • $\begingroup$ I think this is what I am asked for yes. $\endgroup$ – user120768 Apr 18 '15 at 15:18
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$ f(x) = \frac {x^2-y^2}{x^2+y^2} $ is not continuous at $(0,0) $ ,

Because if we approach through real axis $ f(x,0)=1 $ and through imaginary axis $f(0,y)=-1$

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  • $\begingroup$ yes but would it be continuous if a =0? since f(x,y) is not continuous when x^2+y^2=0 $\endgroup$ – user120768 Apr 18 '15 at 15:26
  • $\begingroup$ Whatever be the value of 'a' the function is not going to become continuous because $ L_1 \neq L_2 $ $\endgroup$ – user229886 Apr 18 '15 at 15:30
  • $\begingroup$ ok then thanks :) do you think the first one is correet please? $\endgroup$ – user120768 Apr 18 '15 at 15:33
  • $\begingroup$ I don't think so ! $\endgroup$ – user229886 Apr 18 '15 at 15:34
  • $\begingroup$ do you know how it should be worked out then? $\endgroup$ – user120768 Apr 18 '15 at 15:36

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