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I have the following task:

Define $f_n:[-1,1]\to \Bbb R$ by $$f_n(x)=\begin{cases}1 , \text{ for $-1 \leq x \leq -1/n$} \\ -\sin(n\pi x/2) , \text{ for $-1/n \leq x \leq 1/n$}\\-1 , \text{ for $1/n\leq x\leq 1$} \end{cases}$$

So I need to sketch find the pointwise limit of $(f_n)$. And need to deduce if the convergence is uniform. Then, I need to calculate $f'_n$ and find the limit of $f'_n$.

So here's how I go with it:

I sketch the first 3 $f_n$'s, and he's what I see,

That for every consecutive $n$, $f_n$ converges pointwise to the function $f(x)$ which is the line that lies on the y axis from $[-1,1]$.

So, to determine whether the function converges uniformly, I check whether $$||f_n-f||_{\infty}=||0-1||_{\infty}=1$$

Since the supremum of $|f_n|$ on $0$ is $0$ and Supremum of my upper defined function is $1$ on $0$. Is this correct? (So no uniform convergence)

Now to $f'_n$ is:

$$f'_n=\begin{cases}0,\text{ for $-1\leq x \leq -1/n$}\\ n\pi/2\cos(\frac{n\pi x}{2}), \text{ for $-1/n \leq x \leq 1/n$}\\0 ,\text{for $1/n \leq x \leq 1$}\end{cases}$$

So, $f'_n \rightarrow f''(x)$ where $f''(x) : 0\to 0$ as $n\to \infty$?

Please, if possible, tell me whether I have mistakes in my thinking, and correct it. Any help is appreciated!

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I believe that there are some weak spots in your reasoning. Recall that pointwise convergence means fixing $x\in[-1,1]$ and figuring out what happens to the sequence $\{f_n(x)\}_n$ as $n\rightarrow\infty$.

If $x\in[-1,1]$ and $x\neq0$ then, for $n$ large enough $f_n(x)$ will be identically $1$ or $-1$ depending on the sign of $x$ (check this!). I'll leave it to you to check what happens if $x=0$ and to get the pointwise convergence from there (in particular you need to find the pointwise limiting function $f$).

Further, let me give you a hint on the uniform convergence: $f_n$ is continuous in $[-1,1]$ (check this!) and by now you have the explicit form of the pointwise limit $f$. Now, what do we know about uniform convergence of continuous functions? Can this happen in our example?

Finally, for the derivatives, recall the relation between differentiability and continuity.

Hope this helps.

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