Let $\bar S$ be the closure of a set of functions, suppose $f$ is a function in $\bar S$. Then there exists a sequence of function converging to $f$ by definition of closure.
Is the convergence pointwise or uniform? Why?
I think it is uniform, but I am not sure why this is the case. Can someone help?
Closure in what metric? If it's the sup-norm, the convergence is uniform. If it's some other metric, convergence is in the sense of that metric, and might not be uniform (or pointwise either, for that matter). If it's a non-metrizable topology, there might not be such a sequence.
By the way, if $K$ is an uncountable compact metric space, there is no metric on $C(K)$ (the continuous functions on $K$) such that a sequence converges in this metric iff it converges pointwise.