$\lim_{y \rightarrow b} \lim_{x \rightarrow a} f \neq \lim_{(x,y)\rightarrow (a,b)} f \neq \lim_{x \rightarrow a} \lim_{y \rightarrow b} f$ Can someone give me an example to show that in general
$\lim_{y \rightarrow b} \lim_{x \rightarrow a} f(x,y) \neq \lim_{(x,y)\rightarrow (a,b)} f(x,y) \neq \lim_{x \rightarrow a} \lim_{y \rightarrow b} f(x,y) $
I have been able to construct examples when the first and last limit exist but the middle one does not, but I can't find one where the middle limit exists but is not equal to the other two (both of which exist). 
Does the existence of the middle limit as well as the other two imply that they must be equal to the middle one?

This is an example of a function that is discontinuous at $x=0=y$ and has different limits when  that point is approached from different directions.
