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I am wondering if there can be a sequence of functions $f_n: D \rightarrow \mathbb{R}$ where

i) Each$f_n$ is continuous on $D$,

ii) $f_n$ converges pointwise on $D$ to some $F$,

iii) $f_n$ does not converge uniformly to $F$ and

iv) $F$ is continuous on $D$.

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1 Answer 1

up vote 3 down vote accepted

Yes, consider the maps $f_n : (0,1) \rightarrow \mathbb{R}$ defined by $f_n(x) = x^n$. It converges pointwise but not uniformly to the $0$ map, a continuous function.

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Nice thank you very much! – Starlight Mar 4 '13 at 6:31

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