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?