Let $X$ be a compact metric space and let $D\subset X$ be a dense set. Take a sequence of continuous functions $f_n\colon X\to \mathbb R$ such that $(f_n)$ is uniformly bounded and converges to $0$ pointwise on $D$. Of course, $(f_n)$ need not converge to $0$ pointwise on $X$.
What can we say about the set $$\{x\in X\colon f_n(x)\not\to 0\text{ as }n\to \infty\}?$$
Is it closed? Or discrete?
On
The pointwise limit $f$ of the $f_n$ (if it exists) is a Baire-1 function, and this paper shows that a Baire-1 function that is $0$ on a dense subset is $0$ everywhere.
No. Consider $X=\Bbb R$, $D=\Bbb R\setminus \Bbb Q$, and $f_n$ defined by $$\cases{ f_n(\frac1k)=\frac1k &for $k\in\{1,\ldots,n\}$\\f_n(x)=0&when $x$ lies outside all intervals $(\frac1k-\frac1{n^2},\frac1k+\frac1{n^2})$\\f_n\text{ is linear }&on all intervals $(\frac1k-\frac1{n^2},\frac1k]$ as well as $[\frac1k,\frac1k+\frac1{n^2})$} $$
Then $f_n(x)$ does not converge to $0$ for any $x$ of the form $\frac1k, k\in \Bbb N$ but $f_n(0)=0$ for all $n$.
Note for visualisation: the graphs of these functions are made of spikes that are thinner and thinner as $n$ increases, whose tips lie on the line $y=x$.