Transformation from x-y coordinates to polar not continuous at r=0
In a solution of problem: Determine whether or not the following functions are continuous at the origin.
f = r cos(theta) sin 2(theta), so f tends to0 as r tends to 0 is independent of theta. This proves that f(r,theta )is continuous at
r = 0; but since the transformation between (x, y) and (r, theta) is not continuous at
r = 0, this does not prove that f(x, y) is continuous at the origin. For this we observe that
f(x, y)| =mod (2x^2 y/(x^2+y^2))<=mod 2y
that is f(x) = o(x).
Can anyone please explain what is meant by the transformation between (x, y) and (r, theta) is not continuous at r=0?
Thanks in advance to those who wìll find the time to help