I am having trouble seeing a certain claim that Rudin makes in proving his "Global Cauchy's Theorem":
$\textbf{Cauchy's Theorem.}$ Suppose $f$ is holomorphic in $\Omega$, which is an open set in $\mathbb{C}.$ If $\Gamma$ is a cycle (i.e. a sum of closed paths) in $\Omega$ that satisfies $\text{Ind}_\Gamma (\alpha) = 0$ for every $\alpha \in \Omega^c,$ then $f(z) \cdot \text{Ind}_\Gamma(z) = \dfrac{1}{2\pi i} \displaystyle \int_\Gamma \dfrac{f(w)}{w-z} \ dw$ for $z \in \Omega - \Gamma^*.$
(Here $\Gamma^*$ denotes the set of points in $\mathbb{C}$ that lie on $\Gamma$ and for any $\alpha \in \Omega-\Gamma^*, \text{Ind}_\Gamma (\alpha) = \dfrac{1}{2\pi i} \displaystyle \int_\Gamma \frac{dw}{w-\alpha}.$)
To go about this proof, Rudin introduces a function $g(w,z)$ on $\Omega \times \Omega$ defined to be $\frac{f(w) - f(z)}{w-z}$ if $w \not= z$ and $f'(z)$ if $w=z.$ Then, (using the fact that $g$ is continuous), he defines $$h(z) = \frac{1}{2\pi i} \int_\Gamma g(z,w) \ dw \quad (z \in \Omega).$$ Then, we show that $h$ is holomorphic in $\Omega.$
Then, letting $\Omega_1$ denote the set of complex numbers, $z$, for which $\text{Ind}_\Gamma (z) = 0,$ we define $$h_1(z) = \frac{1}{2\pi i} \int_\Gamma \frac{f(w)}{w-z} \ dw \quad (z \in \Omega_1).$$
We note that for $z \in \Omega \cap \Omega_1, h(z) = h_1(z),$ so that $h \equiv h_1$ on $\Omega \cap \Omega_1.$
Now, his next claim is where I am stuck. He claims that: "Hence, there is a function $\varphi \in H(\Omega \cup \Omega_1)$ whose restriction to $\Omega$ is $h$ and whose restriction to $\Omega_1$ is $h_1$."
I do not see how the above claim follows.
Any help is greatly appreciated.
$$ \varphi(z)=\begin{cases}h(z) & z\in\Omega,\\h_1(z) & z\in\Omega_1.\end{cases}. $$ $\varphi$ is well defined because $h$ and $h_1$ coincide on $\Omega\cap\Omega_1$ and is holomorphic on $\Omega\cup\Omega_1$.