Let $S^1$ be unit circle on $C^\star$(punctured complex plane). Consider $S^1\to S^1$ by $z\to z^2$ map. Identify $Z_2$ action as $e:z\to z$ and $g:z\to -z$ action on $S^1$ of domain. Let $U=S^1-\{1\}$ and $V=S^1-\{-1\}$.
"Let $\sigma:U\to S^1$ be local section defined by taking $\sigma(w)$ be the square root of $w$ with $Im(\sigma(w))>0$ while $\tau:V\to S^1$ is defined by taking $\tau(w)$ to be square root of $w$ with $Re(\tau(w))>0$. "...
Local section $s_u:U\to G,s_v:V\to G$ determines trivialization where $G$ is the group acting on principal bundle.
"$s_u(z)=e$ if $Im(z)>0$ and $g$ if $Im(z)<0$."
"$s_v(z)=e$ if $Re(z)>0$ and $g$ if $Re(z)<0$."
$\textbf{Q:}$ Why is $s_u,s_v$ even continuous or they are sections? Note that if there is section $s:U\to S^1$, then trivialization automatically gives a map $s: U\to U\times Z_2\to Z_2$ where $U\times Z_2\to Z_2$ is canonical projection map. The image space is discrete, it had better be connected. The book defines local section of principal fiber bundle $\pi:P\to M$ with group $G$ as a map $\sigma:U\to P$ s.t. $\pi\circ\sigma=Id_U$. I guess continuity is not required? If that is the case, I do not know what to do.
Here is what I thought the trivialization would be. Consider $U$ with $z\in U\to \sqrt{z}\in S^1-\{\pm 1\}$. This defines a section to $U\to e$. The other section $V\to e $ is identified by $z\in V\to \sqrt{z}\in S^1-\{\pm i\}$. Then I can compute the transition functions which is exactly given in the book for $w\in P, g_{uv}(w)=e$ if $Im(w)>0$ and $g$ if $Im(w)<0$. The only issue I have is the section defined above.
From the comments below, consider $U=S^1-\{\pm 1\}$ and $V=S^1-\{\pm i\}$. Writing down the corresponding $s_u:U\to U\times Z_2\to Z_2$ as section and similarly for $s_v:V\to V\times Z_2\to Z_2$, one can get same transition functions prescribed above.
Ref. Gauge Theory and Variational Principles by David Bleecker, 1.1.7 Example
I had a look into Bleeker's book to understand notation.
In Theorem 1.15 he states that for an open $U \subset M$ there is a $1$-$1$-correspendence between sections $\sigma : U \to P$ and local trivializations $T_U : \pi^{-1}(U) \to U \times G$. In the proof he explains that to a section $\sigma$ we associate $T_U$ with $T_U(\sigma(x)g) = (x,g)$.
Moreover, local trivializations are written in the form $T_U(p) = (\pi(p),s_U(p))$ with $s_U : \pi^{-1}(U) \stackrel{T_U}{\rightarrow} U \times G \stackrel{}{\rightarrow} G$. Thus $T_U$ is uniquely determined by the map $s_U$.
For your $U \subset S^1$ we get $\pi^{-1}(U) = \{z \in S^1 \mid z^2 \ne 1 \} = S^1 \setminus \{1,-1\} = U_{+Im} \cup U_{-Im}$, where $U_{\pm Im}$ are the sets of points with $Im(z) > 0$ and $Im(z) < 0$, respectively. Note that $\sigma(U) = U_{+Im}$.
The associated local trivialization is given by $$T_U(\sigma(w)) = (w,e), T_U(-\sigma(w)) = T_U(\sigma(w)g) = (w,g) .$$ That is, for $z \in U_{+Im}$ we have $s_U(z) = e$ and for $z \in U_{-Im}$ we have $s_U(z) = g$.
Similarly, for your $V \subset S^1$ we get $\pi^{-1}(V) = \{z \in S^1 \mid z^2 \ne -1 \} = S^1 \setminus \{i,-i\} = V_{+Re} \cup V_{-Re}$, where $V_{\pm Re}$ are the sets of points with $Re(z) > 0$ and $Re(z) < 0$, respectively. Note that $\tau(V) = V_{+Re}$.
That is, for $z \in V_{+Re}$ we have $s_V(z) = e$ and for $z \in U_{-Re}$ we have $s_V(z) = g$.