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Recall Dini's Theorem:

Let $K$ be a compact metric space. Let $f: K\to\mathbb{R}$ be a continuous function and $f_{n}: K\to\mathbb{R}$, $n\in\mathbb{N}$ be a sequence of continuous functions. If $\{f_{n}\}_{n\in\mathbb{N}}$ converges to $f$ and if $f_{n}(x)\geq f_{n+1}(x)$ for all $x\in K$ and all $n\in\mathbb{N}$, then $\{f_{n}\}_{n\in\mathbb{N}}$ converges uniformly to $f$.

It is known that the hypotheses:

  • $K$ is compact
  • $f$ is continuous
  • $f_{n}(x)$ decreases as $n$ increases

are necessary. See this document.

Me and my friend are looking for an example where:

  • $K$ is compact
  • $f$ is continuous
  • $f_{n}(x)$ decreases as $n$ increases
  • $f_{n}$ are not continuous
  • $f_{n}\to f$ pointwise, but $f_{n}$ does not converge uniformly to $f$

We have been thinking for 2 hours, but to no avail!

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  • 2
    $\begingroup$ See this. $\endgroup$ – David Mitra Jul 10 '14 at 0:33
  • $\begingroup$ @DavidMitra: Thanks David! Your answer in that thread is very helpful. $\endgroup$ – Prism Jul 10 '14 at 0:51
  • $\begingroup$ You're welcome; glad to help. $\endgroup$ – David Mitra Jul 10 '14 at 1:00
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Here is a modified version of David Mitra's answer from this thread.

$$ f_n(x)=\begin{cases}-1 & 0\le x\le 1-{1\over n}\cr 0&1-{1\over n}< x<1\\-1\;&x=1\end{cases} $$

answers the question. The limit function is the constant function $f=-1$.

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