## Thomas' Calculus 13th Edition

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.