Is the function $f(x,y) = \frac{x^2y^2}{x^2+y^2}$ analytic at the origin?
Changing into polar coordinates $x= r\cos\theta, y=r\sin\theta$ gives $f = r^2 \sin^2\theta \cos^2\theta$. Hence, as a function of $r$ and $\theta$, the function is analytic.
However, I am not sure whether this guarantees that $f$ is analytic, as a function of original variables $x$ and $y$. Also, I am not sure how to write $f$ as a power series, i.e., $$f(x,y) = \sum_{n,m\geq 0} c_{nm} x^n y^m,$$ due to the denominator $x^2+y^2$.
What you have done in using polar coordinates is sufficient to show that the function is continuous at the origin.
For the function to be analytic, not only must it be continuous, it must be infinitely differentiable.
If you take enough derivatives, I think that this function is going to break at the origin.