Let . Then f is , in fact , everywhere except at the origin.

Also,

Thus

and , which does not exist.

Since f(x,y) is symmetric in x and y, it follows that and that does not exist.

Edit.I forgot that you also wanted and to be continuous at the origin. I'll leave you to figure out whether or not that is the case for this example. (I think it's okay, but I haven't checked carefully.)