Properties of the principal square root of a complex number

I am studying the principal square root function of complex numbers. On Wikipedia they present a complex number $z$ using polar coordinates as

$$\begin{equation} z = r \mathrm{e}^{i \varphi}, \quad r \ge 0, ~ -\pi < \varphi \le \pi. \end{equation}$$

Further, they define the principal square root of $z$ as

$$\begin{equation} \sqrt{z} = \sqrt{r} \mathrm{e}^{i \varphi/2}. \tag{1} \end{equation}$$

Continuing, it is mentioned that

The principal square root function is thus defined using the

nonpositive real axis as a branch cut. The principal square root
function is holomorphic everywhere except on the set of non-positive
real numbers (on strictly negative reals it isn't even continuous).

I do not understand these two statements. My questions are

1. Why is the principal square root function defined using the nonpositive real axis as a branch cut? It seems to me that for $z = \mathrm{e}^{i \pi}$, we obtain by equation $(1)$ the principal square root $\sqrt{z} = \sqrt{1} \mathrm{e}^{i \pi/2} = i$.


2. Why is the principal square root function not continuous on the negative reals?

Answer

That is a convention that for principal value in general we must use $-\pi<\arg z<\pi$ for other values you can change the branch cut.