Say we have a sequence of functions $(f_n):(0,1)\rightarrow \mathbb{R}$ which convergence point-wise to a function $f$, and converge uniformly to $f$ on every compact sub-interval of $(0,1)$. Is there anything about the sequence of functions $(f_n)$ that is not preserved in the limit that would be if $(f_n)$ converged uniformly to $f$ on all of $(0,1)$?
Edit: I should clarify, I'm also interested in if the limit can be 'passed' through the integral (assuming it exists) $\int f_n$, incase that wasn't clear.