Q:

Continuity at origin

Given $f=\frac{sin^2(x-y)}{|x|+|y|}$ and $f(0,0)=0$
Check for continuity at origin.
I can see that the function takes a value zero when approaching the origin along the x-axis as well as y-axis. But how to prove/disprove it using first principle?

A:

Hint: Note that $|x-y|\le |x|+|y|$. So for $x$ and $y$ near $0$ but not both $0$ we have
$$0\le\frac{|\sin(x-y)|}{|x|+|y|}\le \frac{\sin(|x+y|)}{|x|+|y|}\lt 1.$$

