# Does pointwise converge imply uniform convergence in some interval?

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.

• The actual theorem along these lines is called Egorov's theorem. – 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.
• Interesting! Are you aware of an example when $f_n$ are all continuous? I foolishly forgot to include that in my original question. – Zachary Hunter Feb 16 at 18:17
• @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. – Ian Feb 16 at 20:01
• @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. – Eric Towers Feb 17 at 3:08