Let $\phi_n\colon [a,b]\to \mathbb R$ be a sequence of continuous functions. Assuming there is an $A\subset [a,b]$ such that $\phi_n|_A$ converges uniformly to $\phi\colon A\to \mathbb R$, I have already proven that for all $x\in \bar A$ the closure of $A$ the sequence $(\phi_n(x))$ converges. Defining $\bar\phi$ on $\bar A$ by the pointwise limit of $\phi_n$ it seems to me, $\bar\phi$ should be continuous on $\bar A$ but I cannot bring a proof together.
My idea: I have already proven, that for any finite set $B\subset \bar A$ the function $\bar\phi$ is continuous on $B\cup A$, and I was now trying to extend this to the entire $\bar A$ with Zorn's Lemma, but showing that, if $A\subset B_1\subset B_2 \subset \dots\subset \bar A$ and $\bar\phi$ is continuous on each $B_k$ then it is also on $\bigcup B_k$, is making some problems. Note that could also assume uniform continuity everywhere.
Note the thoughts of Continuity on a union.