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If I have a sequence of functions $(f_n)$ that converges to a function $f$ that is continuous (and therefore uniformly continuous on compact sets) can I then say that $(f_n)$ converges uniformly to $f$ on compact sets?

I only know that, if the $f_n$ are continuous and convergent uniformly to $f$ that then $f$ is continuous, too.

Rudy the Reindeer
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mathfemi
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    No. Take $f_n(x)=1$ for $0<x<1/n$; $f(x)=0$ otherwise. Then $(f_n)$ converges non-uniformly to the zero function on $[0,1]$. – David Mitra May 27 '14 at 10:34
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    More interesting: there is a sequence of continuous functions with a continuous pointwise limit such that the convergence is not uniform on any open set. See this. – David Mitra May 27 '14 at 10:47
  • Depending on circumstances Dini's theorem, see here: http://en.wikipedia.org/wiki/Dini_theorem, could be of help. – Christian Blatter May 28 '14 at 15:13

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