TITLE: Continuity of $f$ on $\mathbb R^2$
QUESTION [5 upvotes]: The question says: 
Let $$f(x,y) = \begin{cases} \dfrac{\text{sin}^2(x-y)}{|x|+|y|} & \text{if $|x|+|y|>0$} \\ 0 & \text{if $|x|+|y|=0$} \end{cases}$$
Is $f$ continuous on $\mathbb R^2$?  
I just can't get how to prove continuity.can anyone just give a hint for this

REPLY [3 votes]: $$\left|\frac{\sin(x-y)}{|x|+|y|}\right|\le\frac{|x-y|}{|x|+|y|}\le1$$
so that
$$\left|\frac{\sin^2(x-y)}{|x|+|y|}\right|\le|\sin(x-y)|\xrightarrow[(x,y)\to(0,0)]{}0$$
and the function's indeed continous at the origin