Is it true that if $f_n$ converges to $f$ (not necessarily uniformly), and $f$ is continuous, but $f_n$ is discontinuous, then this disproves uniform convergence?

  • $\begingroup$ I'm not sure I understand the question, but the uniform limit of discontinuous functions can be uniformly continuous: $f_n(x) = \begin{cases} 1/n & x = 1/n \\ 0 & x\neq 1/n\end{cases}$ $\endgroup$ – David Bowman Dec 13 '17 at 5:20
  • $\begingroup$ My question rephrased is this. If i have a sequence of discontinuous functions which converge to some continuous function, does this disprove uniform convergence? $\endgroup$ – GTOgod Dec 13 '17 at 5:21
  • $\begingroup$ Yes, as both counterexamples above and below this comment show. $\endgroup$ – астон вілла олоф мэллбэрг Dec 13 '17 at 5:22

Consider $f_{n}(x)=\chi_{[1,n]}(x)e^{-x}$ and $f(x)=e^{-x}$ on $[1,\infty)$, so $f(x)-f_{n}(x)=\chi_{(n,\infty)}(x)e^{-x}\leq e^{-n}$ so $f_{n}\rightarrow f$ uniformly, but $f_{n}$ is discontinuous (at $x=n$), and $f$ is uniformly continuous.

  • $\begingroup$ Sorry, I edited the question. My question was not whether the function f can be uniformly continuous, but whether the convergence of the $f_n$ can be uniform $\endgroup$ – GTOgod Dec 13 '17 at 5:30
  • $\begingroup$ Yes, that's what you want here: $f_{n}$ are discontinuous, $f$ is continuous (uniformly continuous) and the convergence is still uniform, so your assertion does not disprove the possibility of being uniform convergent. $\endgroup$ – user284331 Dec 13 '17 at 5:32

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