Why if a function is holomorphic and injective in neighbourhood of $x_0$ then $f'(x)\ne 0$ in neighbourhood of $x_0$? Why if a function is complex differentiable and injective in some neighborhood of $x_0$ then its derivative is non zero in that neighborhood?
I just don't see how why it is like that. Obviously in real case it's not true.
 A: If holomorphic function $f$ is not constant and $f'(p) = 0$, then $p$ is a zero of $f'$ of some multiplicity $m \ge 1$, and a zero of $f - f(p)$ of multiplicity $m+1$.  Suppose the closed disk $D_r(p)$ of radius $r > 0$ centred at $p$ is contained in the neighbourhood where $f$ is holomorphic, and $r$ is small enough that $D$ contains no other zeros of $f'$.  By the Argument Principle, the number of zeros of $f(z) - t$, counted by multiplicity, in the interior of $D_r(p)$ is constant for $t$ in a neighbourhood of $f(p)$.  But since the zeros of $f'$ are isolated, none
of these zeros of $f(z)-t$ can have multiplicity $> 1$.  Thus for $t \ne f(p)$ in this neighbourhood, there is more than one $z$ with $f(z)=t$.  
A: Let us assume that $f'(x_0) = 0$ and let $V$ be a neighborhood of $x_0$ in which $f$ is holomorphic. We can find (by analycity of $f$) an integer $k \geq 2$ and we can find (by local inversion) a conformal mapping $g$ from a neighborhood $W \subset V$ of $x_0$ to a disk $D(0,\epsilon)$ with $\epsilon > 0$ such that $f(z) - f(x_0) = g(z)^k$ for all $z \in W$.
Therefore, for all $w \in B(0,\epsilon)$, the equation $f(z) - f(x_0) = w$ is equivalent to the equation $g(z)^k = w$ for $z \in W$, hence it has exactly $k$ distinct solutions in $W$.
In conclusion, it is necessary that $f'(x_0) \neq 0$ for $f$ to be injective in $W$. Moreover, $f'(z) \neq 0$ in a neighborhood of $x_0$ by continuity of $f'$. 
A: if $f'(x_0) = 0$, then $f(x)$ is not injective in a sufficiently small neighborhood of $x_0$. Using the Taylor series expansion
$$f(z) \approx a_0 + a_2(z - x_0)^2 ...$$
it is clear that $f(z)$ will take on values sufficiently near $a_0$ at two points in a sufficiently small neighborhood of $x_0$.
