Integral of a f(x,y)=xy over the circle with radius 2 oriented positively (Complex Analysis)

Question

I have to integrate $f(x,y) = xy$ over $|z| = 2$. I've worked out the integral using the fact that $x = \frac{z+\bar{z}}{2}$ and $y = \frac{z-\bar{z}}{2i}$ ending up with $f(z) = \frac{-i(z^2-\bar{z}^2)}{4}$ Now I define the parametrization $\gamma(t) = 2e^{it}$ and finally I find that the value of this integral is zero. Here comes my question, shouldn't my integral be something different than zero? Because using the C-R equations I find that $f$ is only analytic at $x = 0$ and $y = 0$ and from what I've read and understand if $f$ is not analytic at every point on the inside of $\gamma$ then the integral should be different than zero. Can someone please help me with this question please? Edit: Or I mean that If $f$ is analytic at all points interior to and on $\gamma$ then the integral over $\gamma$ is zero. But in this case I don't have that :( I'm quite confused.