Exercise 6.3.6

Question

Provide an example or explain why the request is impossible. Let's take the domain of the functions to be all of $\mathbf{R}$.
  \begin{enumerate}[label=(\alph*)] 
  \item A sequence $\left(f_{n}\right)$ of nowhere differentiable functions with $f_{n} \rightarrow f$ uniformly and $f$ everywhere differentiable.
  \item A sequence $\left(f_{n}\right)$ of differentiable functions such that $\left(f_{n}^{\prime}\right)$ converges uniformly but the original sequence $\left(f_{n}\right)$ does not converge for any $x \in \mathbf{R}$.
  \item A sequence $\left(f_{n}\right)$ of differentiable functions such that both $\left(f_{n}\right)$ and $\left(f_{n}^{\prime}\right)$ converge uniformly but $f=\lim f_{n}$ is not differentiable at some point.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
    \item Let $g(x)$ be the continuous but nowhere-differentiable function defined in section 5.4. Then since $g(x)$ is bounded, $f_n(x) = g(x) / n$ clearly converges uniformly to 0, and is thus such a sequence.
    \item Let
    $$f_n(x) = \begin{cases}
       1 & \text{  n is odd} \
       0 & \text{  n is even} \
    \end{cases}$$
    Clearly $f_n$ does not converge anyewhere, but $f'_n(x)$ converges to 0.

\item Not possible - since $f_n$ converges uniformly, it must converge at at least one point, and since $f'_n$ converges uniformly, we can apply Theorem 6.3.3 to show that $f$ is differentiable everywhere.
 \end{enumerate}

Exercise 6.3.6

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Exercise 6.3.6

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