Let $X$ be a metric space and $\{f_n\}$ be a sequence of functions such that $f_n:E\rightarrow X$.
Suppose $f_n\rightarrow f$ uniformly on a set $E$ and $x$ is a limit point of $E$ and $\lim_{t\to x} f_n(t)=A_n$. ($E$ is a subset of some metric space $Y$.)
I know that if $X$ is complete, then $\{A_n\}$ converges. However, i'm really not sure if it is essential. Is it?