Why does the principle value of $z^{\frac{1}{2}} := \sqrt{r}e^{i\frac{\theta}{2}}$ need to be restricted to one rotation in order to be continuous? I understand that the function $f(z) = z^{\frac{1}{2}}$ is multi-valued, since it has exactly two solutions for every $z$. However if I take the principle value square root function, expressed as
$$f(r,\theta) = \sqrt{r}e^{i\frac{\theta}{2}}$$
Then why does a branch cut need to be taken here? From what I've read, if I want this function to be continuous then I should restrict $\theta$. But as far as I understand, $e^z$ is single-valued for any complex number $z$. 

$f(r,\theta)$ is continuous, but the map $z\mapsto(r,\theta)$ is multi-valued, due to $Arg(z)$ is multivalued. If you want $z\mapsto(r,\theta)$ to be continuous, you have to select a single valued branch and restrict the range of $\theta$.
