I search a lot for the answear and many topics here about this questions says the function is continuous on $[0,\infty)$. But the book doesn't says anything about this interval. It requests to prove for all set of complex. Well, what I know is for any point $z$ and $z_0 \neq 0$ we have
$|f(z) - f(z_0)| = |\sqrt{z}- \sqrt{z_0}|= | \frac{z-z_0}{\sqrt{z}+\sqrt{z_0}} | < \epsilon$.
Now what should I do to prove this using $\sqrt{z} = r^{\frac{1}{2}}e^\frac{i\theta}{2}$ and $\sqrt{z_0} = r_0^{\frac{1}{2}}e^\frac{i\theta_0}{2}$ ?
The book says to consider the fact $|\sqrt{z}+\sqrt{z_0}|^2= r + r_0 + 2\sqrt{rr_0}cos(\frac{\theta + \theta_0}{2}) > r_0 $
It isn't continuous on all of $\Bbb C$. You need a branch of the square root function, and the largest subset you can define it on is $\Bbb C$ minus a slit, like $(-\infty,0)$.