# Difficult Continuity Multivariable Question

WIll you help me understand the following?

$f(x,y)=\begin{cases} \sin(y-x) & \text{for} & y>|x| \\ \\ 0 & \text{for} & y=|x| \\ \\ \frac{x-y}{\sqrt{x^2 + y^2}} & \text{for} & y<|x| \end{cases}$

I need to check differentiability and continuity. I tried substituting $x= \frac{1}{2} (u-v) , y=\frac{1}{2} (u+v)$ but it doesn't help me... Will you help me figure this thing out?

Thanks !

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Hint:
Calculate $\lim\limits_{x\to{0}}{f(x,\,y)}$ along lines $y=kx$ for different $k.$

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Great ! THanks ! . So the limit does not exist at all ! (right?) –  czash May 28 '13 at 5:17
Yes, the limit does not exist. –  M. Strochyk May 28 '13 at 5:43