Let $f_n$ be sequence of continuous functions which converges pointwise to $f$ on a closed interval, $E$, must there be closed interval $E' \subseteq E$ such that $f_n|_{E'}$ converges uniformly to $f|_{E'}$.
This is a modified version of this question, where I unintentionally forgot to specify that $f_n$ is continuous.
This answer to a related question gives a construction of a sequence of continuous functions defined on $[0,1]$ converging pointwise to a function whose discontinuities are dense in $[0,1]$. In particular, the convergence cannot be uniform on any subinterval of $[0,1]$, since then the limiting function would be continuous on that subinterval.
The other answers, comments, and links from that question should be interesting reading for you; this is the type of question that has been well studied.
