If we say $$(f_n)_{n = 1}^\infty \ a \ sequence \ of \ functions\ f_n: D \to \mathbb{C}\ \ converge\ uniformly\ \ on \ f: D \to \mathbb{C}$$
For any subset $$ S \subseteq D$$ , is it tru that for the functions $$f_n: S \to \mathbb{C} \ and f: S \to \mathbb{C}\ have \ f_n \to f$$ uniformly on S ?
In other words, if we restrict the domain of the function that is uniformly convergent , does it remain uniformly convergent ?
Yes. Recall the definition. Uniform convergence in $D$ means the following:
$\forall(\epsilon>0) \exists (n_0\in\mathbb{N})\ \forall (n\geq n_0)\ \forall (z\in D)\ [|f_n(z)-f(z)|<\epsilon]$
So if the condition in the end holds for each $z\in D$ then in particular it holds for all $z\in S$ when $S\subseteq D$. For a given $\epsilon>0$, the same index $n_0$ will work. Thus $f_n\to f$ uniformly in $S$ as well.