Given the function
$ f(z) = \frac{x+iy}{x^2+y^2} $
I need to determine whether the function is analytic.
Using Cauchy-Riemann equation,
$ u_x = \frac{y^2-x^2}{(x^2+y^2)^2} $
$ v_y = \frac{x^2-y^2}{(x^2+y^2)^2} $
The solution's manual states that this function is analytic only at the origin because $u_x = v_y$ is satisfied at the origin. My question is, aren't $u_x$ and $v_y$ undefined at $x = y = 0$ due to division by zero?