Example for a sequence of continuous functions which converge to a continuous function pointwise but not uniformly on a compact set.

Question

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.

Answer 1

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.

Answer 2

Answer based on hints given by Nate Eldredge. But this one I obtained from one more book.

$f_{n}(x)=n^{2}x$ if $ 0 \leq x \leq \frac{1}{n} $,

= $-n^{2}x+2n$ if $ \frac{1}{n} <x< \frac{2}{n}$

=$0$ if $\frac{2}{n} \leq x \leq 1$ .

Here we have to start with $n=2$.

Graph of this function will look like triangle with changing base.