A function is "continuous", of course, if and only if the limit exists and is equal to the value of the function at the point. Here the value of the function is always 0 so the problem is to determine whether the limit exists and is 0.
In order that a limit exist, the value of the function as you approach (0, 0) along any path must be the same so if youtry two different paths and get two different limits, the limit does NOT exist and the function is NOT continuous.
But you can never try "all" paths so that will never tell you that the limit does exist, only when it does not. I would recommend changing to polar coordinates. That way, the "r" variable alone measures the distance to the origin. If the limit, as r goes to 0, exists no matter what is, then the limit, as (x, y) goes to (0, 0), exists and is that value.