My actual aim is to verify that each of the conditions in Dini's theorem is essential or not.Theorem says that A sequence {$f_{n}$} of continuous functions defined on a compact set $E$ converges to $f$ point wise satisfying the following properties :
$f_{n}$ is monotonic
and $f$ is continuous.
Then {$f_{n}$} converges to $f$ uniformly.
So there are three hypothesis are used in theorem compactness of $E$,continuity of $f$ and monotonicity of $f_{n}$.
First one is essential because $f_{n}= \frac{1}{1+nx}$ defined on open interval $(0,1)$ converge to $o$ point wise but not uniformly.Second condition cannot be omitted since $f_{n}=x^{n}$ defined on $[0,1]$ converges to a discontinuous function.
Left out one is monotonicity.For that I need to construct a sequence of functions which are continuous, non-monotonic,and converges point wise to a continuous function.But convergence should not be uniform.
I could not get such a sequence.
Best Answer
Let $f_n$ be a big "spike" with support in $[0, 2/n]$. To be precise:
For $n \geq 2$, let $f_n$ be the unique piecewise linear function on $[0,1]$ such that:
(a) $f_n = 0$ on $[2/n, 1]$,
(b) $f_n$ is linear on $[0,1/n]$ and $[1/n, 2/n]$,
(c) $f_n(0) = f_n(2/n) = 0$, and
(d) $f_n(1/n) = n$.
The $f_n$ converge to zero, but certainly not uniformly.