Multiple-valued analytic functions 
Although our definition requires all analytic functions to be single-valued, it is possible to consider such multiple-valued functions as $\sqrt{z}$, $\log z$, or $\arccos z$, provided that they are restricted to a definite region in which it is possible to select a single-valued and analytic branch of the function.
For instance, we may choose for $\Omega$ the complement of the negative real axis $z\le 0$; this set is indeed open and connected. In $\Omega$ one and only one of the values of $\sqrt{z}$ has a positive real part. With this choice $w=\sqrt{z}$ becomes a single-valued function in $\Omega$; let us prove that it is continuous.

(The set $\Omega$ is the open set on which $f$ is defined.)
I don't really understand what all this means. Why is the negative real axis described by $z\le 0$ (shouldn't it be $x\le 0$?) Isn't it always the case that one value of $\sqrt{z}$ has a positive real part, since the two values are negative of each other? And why does $w=\sqrt{z}$ become a single-valued function, when we restrict the domain but not the range?
 A: 
Why is the negative real axis described by $z\le  0$ (shouldn't it be $x\le 0$?)

Whenever a complex number appears in an inequality, the inequality implicitly says that the number is real. Hence, $z\le 0$ is indeed the negative real axis. The inequality $x\le 0$, on the other hand, would be understood as  describing the left half-plane (since it does not say anything about $y$). 

Isn't it always the case that one value of $\sqrt{z}$ has a positive real part

True (if "positive" means $\ge 0$). But after removing the negative semiaxes we can say the same, but with "positive" being $>0$. The difference is substantial. Strict inequalities are stable under small perturbations; thus, choosing the root with positive real part gives us a   continuous function. 

And why does $w=\sqrt{z}$ become a single-valued function, when we restrict the domain but not the range?

The range is determined by the domain. Whenever  we talk about the restriction of some function, it's the domain that is being restricted. 
There are still two values of square root to choose from, but it's now possible to make the choice continuously,  (and, consequently, in a holomorphic way). 
