$f_n,f_n^{\prime}\rightarrow 0$ pointwise. But $(f_n^{\prime}) $ is not uniformly convergent. Find an example of such a sequence of functions.

Question

I was studying uniform convergence of sequence and series. I encounter this problem. Trying to find an example. Any help will be appreciated. The domain of the function can be any closed interval. For example, [0,1] will suffice.

I am looking for an example to show that the conditions of the following theorems are necessary but not sufficient:

Let $f_n\colon [a,b]\rightarrow \mathbb{R}$ be a sequence of differentiable functions on $[a,b]$, and let $f_n\rightarrow f$ pointwise on $[a,b]$. If $f_n^{\prime}\rightarrow g$ uniformly on $[a,b]$, then $f$ is differentiable on $[a,b]$ and $f^{\prime} = g$.

score 1 · Accepted Answer · answered Aug 01 '16 at 01:43

Let $f_n(x) =x^{n+1}/(n+1)$ on $[0,1].$ Then $f_n \to 0$ uniformly on $[0,1,],$ and $f_n'(x) =x^n$ converges to $\chi_{\{1\}}$ pointwise on $[0,1].$ The convergence of $f_n'$ can not be uniform on $[0,1],$ because each $f_n'$ is continuous on $[0,1]$ while the pointwise limit of $f_n'$ is not continuous there.

$f_n,f_n^{\prime}\rightarrow 0$ pointwise. But $(f_n^{\prime}) $ is not uniformly convergent. Find an example of such a sequence of functions.

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