I know that uniform convergence of a sequence of continuous functions $f_n$ on some set $S$ implies continuity of the limit function $f$ on that set. I also know that pointwise convergence of the sequence of continuous functions $f_n$ on some set $S$ does not imply continuity of the limit function $f$ on that set $S$ because if only pointwise convergence is assumed the limit function can be discontinuous.
Now, the natural question that arises is the following:
What is the weakest known condition on the sequence of continuous functions $f_n$ defined on the set $S$ (if there is additional condition imposed on the set $S$ please, state it, although I believe that such condition should be imposed only on the sequence of functions if the set is nice enough, so you can take the set $S$ to be open or compact) such that that condition guarantees(implies) continuity of the limit function $f$ on the set $S$?
EDIT:(14.10.2015) I have the following question which I will ask here so that I do not pose another question as the question that I am going to ask is very closely related to the question asked above and the question is:
Suppose that $f_n$; $n\in \mathbb{N}$, is a sequence of continuous functions defined on some set $S$ (you can take $S$ to be as nice as possible but as general as possible (open, compact)). What is the weakest known condition on the sequence $f_n$ such that the limit function $f$ is continuous on $S$ if and only if that condition is satisfied?