I noted the answers in this question, but the accepted answer does not seem to me to provide a correct approach in showing that a condition is essential in a theorem as it has proven the theorem and not provided any examples which do not satisfy the mentioned condition i.e (continuity of each $\{f_n\}$). Thus, I pose this question here. I am trying to show that the continuity of each $\{f_n\}$ is essential in Dini's theorem. Let me state the theorem for ease of access first:
If $\{f_n\}$ is a sequence of real-valued continuous functions converging pointwise to a continuous limit function $f$ on a compact set $S$, and if $f_n(x)\geq f_{n+1}(x)$ for each $x$ in $S$ and every $n=1,2,\dots$, then $f_n\to f$ uniformly on $S$.
I already have shown that the compactness condition is essential by considering $f_{n}(x)=x^{n}(1-x)$ on the interval $(0,1)$, yet in order to show that the continuity of each $f_{n}$ is essential, I have not been able to provide an example in which only the continuity of each $f_{n}$ is not satisfied yet all the others are. Could you furnish me with an example? Any help of yours is appreciated.
