I was wondering if $f_n$ converges pointwise to $f$ on a closed interval, $E$, must there be closed interval $E' \subseteq E$ such that $f_n|_{E'}$ converges uniformly to $f|_{E'}$.

Currently, with my intuition, I can't think of an counter-example.

  • $\begingroup$ The actual theorem along these lines is called Egorov's theorem. $\endgroup$ – Ian Feb 16 at 17:54

Let $(q_n)_{n\in\mathbb N}$ be an enumeration of $\mathbb Q\cap[0,1]$. For each $n\in\mathbb N$, let $f_n\colon[0,1]\longrightarrow\mathbb R$ be equal to $\chi_{\{q_n\}}$; in other words, $f_n(x)=0$, unless $x=q_n$, in which case $f_n(x)=1$. Then $(f_n)_{n\in\mathbb N}$ converges pointwise to the null function. However, there is no interval $[a,b]\subset[0,1]$ such that the convergence is uniform on $[a,b]$, since $[a,b]$ contains infinitely many $q_n$'s.

  • $\begingroup$ Interesting! Are you aware of an example when $f_n$ are all continuous? I foolishly forgot to include that in my original question. $\endgroup$ – Zachary Hunter Feb 16 at 18:17
  • $\begingroup$ @ZacharyHunter I'm pretty sure you can just imitate this example in the continuous case: take $f_{m,n}$ continuous with $\lim_{m \to \infty} f_{m,n} = \chi_{\{ q_n \}}$ pointwise, then enumerate that 2D array into a sequence. $\endgroup$ – Ian Feb 16 at 20:01
  • $\begingroup$ @ZacharyHunter : Just put a bump of small width, say $n^{-2}$ centered at $q_n$ in $f_n$. This is a generic method of making a sequence of point-indicator-like functions into a sequence of continuous functions. $\endgroup$ – Eric Towers Feb 17 at 3:08

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