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Consider $f_\nu:\Omega\subset\Bbb R^n\mapsto \Bbb R,\ \nu\in\Bbb N$ some functions

Local uniform convergence is when $\forall x\in\Omega$ there exists an open neighborhood $U_x\ni x$ of $\Omega$ such that $f$ converges uniformally in $U_x$

And my question is: does pointwise convergence imply local uniform convergence? (I'm pretty sure local uniform $\implies$ pointwise, that would make those two equivalent, but then what's the point of defining the local uniform convergence...)

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  • $\begingroup$ Hint: Local uniform limit of continuous functions is continuous again. $\endgroup$
    – user251257
    May 7 '18 at 17:48
  • $\begingroup$ $f_\nu$ are not necessarily continuous, or is that irrelevant? $\endgroup$ May 7 '18 at 18:05
  • $\begingroup$ it is just an easy way to found a counter example. $\endgroup$
    – user251257
    May 7 '18 at 18:09
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Consider $f_n(x) = x^n$ on $[0,1]$, which converges pointwise.

However, for any neighborhood of $x=1$, there is another point $x'$ such that $|f_n(x') - 0|$ is close to $1$.

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