Answer
No.
Work Step by Step
Consider two difference approaches to (0,0):
1. along y=-x, we have $\lim\limits_{x,y \to 0,0}\frac{sin(x-y)}{|x|+|y|}=\lim\limits_{x,y \to 0,0}\frac{sin(2x)}{2|x|}=1$ for $x\gt0$
2. along y=2x, we have $\lim\limits_{x,y \to 0,0}\frac{sin(x-y)}{|x|+|y|}=\lim\limits_{x,y \to 0,0}\frac{sin(-x)}{3|x|}=-\frac{1}{3}$ for $x\gt0$
Hence, function $f(x,y)$ is not continuous at the origin.
