# finding the continuity of a function

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 ?

• Your "second part" is not clear. Do you mean making the function $f$ continuous in both variables at all points? – Rory Daulton Apr 18 '15 at 15:16
• I think this is what I am asked for yes. – user120768 Apr 18 '15 at 15:18

## 1 Answer

$f(x) = \frac {x^2-y^2}{x^2+y^2}$ is not continuous at $(0,0)$ ,

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

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