You mean here, don't you? That's very different from saying "the derivative exists at (0, 0)" or "the function is differentiable at (0, 0)"

Yes, thi partial derivative exists at x= 0 if and only if exists.

In this case, that limit is . It does NOT look to me like that limit exists. Are you sure you have copied the problem correctly?

Then, I can imagine it is discontinuous because at (0,0) the graph of should be broken, which has the circle there, but I just forgot how to show it. I thought the definition of "continuiy" is the value of limit equals to the value of function. So how do I write about it? Please help me, thanks a lot.