A counter-example to Dini's Theorem (after removing a hypothesis)

Question

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!

Answer 1

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 $f=-1$.