This statement of the theorem might be helpful to you: www.math.uiuc.edu/~tyson/existence.pdf. (For f and its partials you are considering a selection of inputs which you are interested in.)
The function f which you gave is not continuous at (0, y) because it is not defined there. Since f is a rational function, it is continuous on its domain, so, for x and y any reals except x=0.
For a function of 2 variables to be continuous at a point, you must have:
f(a,b) exists, and \[\lim_{(x,y) \rightarrow (a,b)} f(x,y)\] exists, and \[\lim_{(x,y) \rightarrow (a,b)} f(x,y) = f(a,b)\] (This must hold as the (x,y) approaches (a,b) along any direction or path.)