Does locally uniform convergence on a dense set imply pointwise convergence? Assume $x^n$ and $x$ are continuous on $[0,T]$ and $A$ is a dense set of $[0,T]$. Moreover, $x^n$ converges to $x$ (as $n$ goes to infinity) locally uniformly in $A$, i.e., for each $t\in A$, we have
$$\lim_{\delta \to 0}\lim_{n\to \infty}\sup_{t-\delta\le s_1,s_2\le t+\delta}|x^n_{s_1}-x_{s_2}|=0.$$
Do we have $x^n_t\to x_t$, for each $t\in [0,T]$ (pointwise convergence)? If not, could anyone have a counterexample? 
Thanks a lot!

Counterexample: Let $T=1, A = [0,1), x\equiv 0, x_n(t) = t^n .$
