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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.

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    Maybe think about the functions $f_n(x) = x^{-1/n}$. The sequence $f_n(x)$ converges uniformly to $1$ on every compact subinterval of $(0,1)$, thus the limit clearly doesn't see the behavior of $f_n$ near $0$.2012-03-06
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    Just to be clear: are you assuming the functions are integrable on $(0,1)$? --- Okay, I see. Perhaps not.2012-03-06
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    Well, you can, what I'm interested in is if you do assume they are, does the limit pass through the integral?2012-03-06

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