This mapping is discontinuous at (0,0).
Therefore, since you include this point in your mapping, it is discontinuous.
Remark that if your mapping would be then it would be continuous.
Now to prove that the function is discontinuous at (0,0), you need to show that
We don't care wath function f is since we can prove that the limit doesn't exist.
Let approach the point along the line of equation The limit then become Since the limit depends on m, by uniqueness of limit we conclude that the limit doesn't exist.
Finally, is not continuous since is not.