Here's a thought - but note that I am not as smart as you!

I think he is trying to say that there is "trouble" at all points of the function f(0,y), not just the point where y=0. So, I think you should have shown that the function is continuous at all points (0,y), and not just the point (0,0).

Here is what you did - you showed that the function was continuous for all points (x,y) where x is not 0, and then you showed that the function was continuous at the origin. This leaves many points of the form (0,y) where y is not 0.

For example, is f continuous at (0,2)?