. . . . . . . . . . .
In that definition, x is a variable point in S, and the absolute value sign denotes the metric in S (which is presumably meant to be a metric space).
If you let n → ∞ in that definition, and if f_n → f pointwise, then you see that . That shows that f is continuous. It need not be true that f is uniformly continuous. For example, if S=(0,1) and each f_n is the function 1/x then obviously f is also the function 1/x.