Not clear whether ratio of two holomorphic functions is holomorphic function except points whether denominator is zero By the link Proof that a function is holomorphic it is told

Elementary operations or compositions of holomorphic functions give holomorphic functions on the maximal domain where the functions are defined. This is a consequence of the rules of derivation for product, ratio and compositions of functions.

But per my understand I have counterexample
$$
f(z) = f(x + \imath y) = \frac{x - \imath y}{x + \imath y}.
$$
I calculate by dividing complex that
$$
f(x + \imath y) = 1 - 2\imath\frac{xy}{x^2 + y^2} = u(x, y) + \imath v(x, y).
$$
Then I verify Cauchy-Riemann criteria
$$
\frac{\partial u}{\partial x} = 0,
$$
and while this
$$
\frac{\partial v}{\partial y} = -2x\frac{x^2 - y^2}{(x^2 + y^2)^2},
$$
which means that $f$ is not holomorphic.
Did I made mistake in calculations or this means that cited statement is not correct?

Take the representation $z=x+y i$. Then the function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined by $f(z)=\overline{z}=x-y i$ is not a holomorphic function. It is antiholomorphic. See http://en.wikipedia.org/wiki/Complex_conjugate
