Consider the function $$ f(x,y) = \begin{cases} \displaystyle{\frac{\sin (x-y)}{|x|+|y|}} &\mbox{ if } (x,y)\not=(0,0), \\ 0 &\mbox{ if } (x,y)=(0,0). \end{cases} $$
I was asked to check the continuity of the function at the origin, so I think I need to use method of polar coordinate, where $x= r \sin \theta$ and $y = r \cos \theta$, and substitute it into the equation.
But after trying to simplify it, I found that the $r$ in the bottom cannot be cancelled out, and I know $r$ will tends to $0$ as $(x, y)$ tends to $0$. So $r$ must be cancelled out in the denominator.
Is there some trigonometric functions that can be applied so that the $r$ in the nominator can be taken out as common factor and is cancelled out with the $r$ in the denominator?
The function $f$ is not continuous at the origin. Consider the path $y=0$. Then $$ \lim_{x\rightarrow 0^+ } \frac{\sin x}{|x|} =\lim_{x\rightarrow 0^+ } \frac{\sin x}{x} = 1 $$ while $$ \lim_{x\rightarrow 0^- } \frac{\sin x}{|x|} = \lim_{x\rightarrow 0^- } \frac{\sin x}{-x} = -1. $$