Is $||x|-|y||-|x|-|y|$ continuous everywhere?

Question

The question asks for me to check the continuity of the function $$f(x,y)=||x|-|y||-|x|-|y|$$ for $(x,y) \in \mathbf{R}^2$ . For points with $|x|>|y|$ it is $-2|y|$, for points with $|x|<|y|$ it is $-2|x|$, and for points with $|x|=|y|$ it is $-(|x|+|y|)=-2|x|=-2|y|$. So the function seems to be continuous everywhere. Is there anything else I am missing?