I'm reading a proof of Arzela - Ascoli theorem (complex analysis version) and I've come to the part where we want to extend a function that is defined on one set to the closure of that set. The problem is that it's stated that "the function is continuous on every compact set in our domain, therefore we can extend it continuously on the closure" (our domain is open and connected set in $\mathbb{C}$). I'm quite sure that the idea should be to prove uniform continuity on the whole set and then extend it, and this is a bit harder.
Definitions:
Let $\Omega \subset \mathbb{C}$ be an open and connected set.
- Family of functions $ \mathfrak{F}=\{f_i : \Omega \to \mathbb{C} | i\in I\}$ is normal iff for every sequence $(f_n)_{n \in \mathbb{N}}$ in $\mathfrak{F}$ there is a subsequence $(f_{n_k})_{k \in \mathbb{N}}$ and there is $f\in \mathfrak{F}$ such that for every compact $K\subset \Omega$, $(f_{n_k})$ converges to $f$ uniformly on $K$.
- Family of functions $ \mathfrak{F}=\{f_i : \Omega \to \mathbb{C} | i\in I\}$ is equicontinuous on $\Omega$ iff $(\forall z_0 \in \Omega)(\forall \epsilon >0)(\exists \delta >0)(\forall f \in \mathfrak{F})(\forall z\in \Omega)(|z-z_0|<\delta \implies |f(z)-f(z_0)|<\epsilon)$
Note that equicontinuity is sometimes (always?) defined as property in which $\delta$ in the expression above is independent from $z_0$. On compact sets these statements are equivalent.
Theorem: Let $ \mathfrak{F}$ be an equicontinuous family on $\Omega$. If $(\forall z\in \Omega) (\{f(z) | f \in \mathfrak{F}\}$ is sequentially compact), then $ \mathfrak{F}$ is a normal family.
Proof of the theorem starts by introducing $\Omega_1 = \{x+iy | x,y \in \mathbb{Q}, x+iy \in \Omega\}$. $\Omega_1$ is dense in $\Omega$ and countable. Since it's countable, it can be written as $\Omega_1=\{z_n|n\in \mathbb{N}\}$.
Then we take a sequence $(f_n)_{n \in \mathbb{N}}$ in $\mathfrak{F}$ and we want to make $f\in \mathfrak{F}$ and $(f_{n_k})_{k \in \mathbb{N}}$ that converges to $f$ on compacts in $\Omega$.
We take $z_1\in \Omega_1$. $\{f(z_1) | f \in \mathfrak{F}\}$ is sequentially compact, so we have a convergent subsequence $(f_{n_k}(z_1))_{k \in \mathbb{N}}$. For first element of our subsequence we take $f_{n_1}$. Then we observe $(f_{n_k}(z_2))_{k \geq 2}$ and repeat the process etc.
We define $f:\Omega_1 \to \mathbb{C}$, $f(z_j)=lim_{k \to \infty}f_{n_k}(z_j)$. We want $f$ to be defined on $\Omega$ and this is where the problem is. For $z,w \in \Omega_1$ $|f(z)-f(w)|\leq |f(z)-f_{n_k}(z)|+|f_{n_k}(z)-f_{n_k}(w)|+|f_{n_k}(w)-f(w)|$.
At this point we somehow start observing all of this on a compact set. I don't know what to do with this.
Edit: the reason why we observe this on a compact set is because we want to prove that $f$ is continuous on every compact set and therefore on $\Omega_1$ and this can be done. However, this doesn't guarantee the existence of the continuous extension. For that we need uniform continuity on the whole domain.