Let $X$ be a Banach space and $H$ be a subset dense in $X $.
Let $\{f_n\}$ be a squence of function from $X$ to $\mathbb{R}$ is equicontinuous on $X$ such that:
$$ \forall x\in H~,~\exists\phi_x\in\mathbb{R}~,~ \phi_x=\lim_{n} f_n(x) $$
Can we say that : $$ \forall y\in X~,~\exists\phi_y\in\mathbb{R}~,~ \phi_y=\lim_{n} f_n(y) $$