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So (correct me if I'm wrong please), this following is a fact:
For a sequence of $\textbf{continuous}$ functions $\{f_n(x)\}_{n=1}^{\infty}$, the series $\sum_{n=1}^{\infty} f_n(x)$ converges uniformly to $f \implies$ $f$ is continuous.

So the negation of that (please correct me if I'm wrong):
$f$ is not continuous $\implies$ the sequence $\{f_n(x)\}_{n=1}^{\infty}$ are $\textbf{discontinuous}$ functions $\textbf{or}$ the series $\sum_{n=1}^{\infty} f_n(x)$ does not converge uniformly.

I wonder if my logical negation in the second step is 100% correect (and if the original statement is true)

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The conclusion should be either the series is not uniformly convergent or that at least one of the $f_n$ is discontinuous.

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  • $\begingroup$ Thank you, so the original statement is also correct? $\endgroup$ – Natash1 Jun 5 '17 at 5:05
  • $\begingroup$ Yes, a uniform limit of continuous functions is continuous. $\endgroup$ – Lord Shark the Unknown Jun 5 '17 at 5:19
  • $\begingroup$ Is there any theorem that says that (this isn't related to the question, but it's a subquestion I guess), if $f_n$ converges pointwise to $f$, and $f_n$ is continuous, then if $f$ is not continuous, $f_n$ mustn't converge uniformly?? $\endgroup$ – Natash1 Jun 5 '17 at 5:23
  • $\begingroup$ Dini's theorem is a partial converse: en.wikipedia.org/wiki/Dini%27s_theorem $\endgroup$ – Lord Shark the Unknown Jun 5 '17 at 5:25

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