Exercise 6.2.2

Question

\begin{enumerate}[label=(\alph*)] 
  \item Define a sequence of functions on $\mathbf{R}$ by
    $$
    f_{n}(x)= \begin{cases}1 & 	ext { if } x=1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n} \ 0 & 	ext { otherwise }\end{cases}
    $$
    and let $f$ be the pointwise limit of $f_{n}$.
    Is each $f_{n}$ continuous at zero? Does $f_{n} ightarrow f$ uniformly on $\mathbf{R} ?$ Is $f$ continuous at zero?
  \item Repeat this exercise using the sequence of functions
    $$
    g_{n}(x)= \begin{cases}x & 	ext { if } x=1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n} \ 0 & 	ext { otherwise. }\end{cases}
    $$
  \item Repeat the exercise once more with the sequence
    $$
    h_{n}(x)= \begin{cases}1 & 	ext { if } x=\frac{1}{n} \ x & 	ext { if } x=1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n-1} \ 0 & 	ext { otherwise. }\end{cases}
    $$
    In each case, explain how the results are consistent with the content of the Continuous Limit Theorem (Theorem 6.2.6).
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Each $f_n$ is continuous at zero, but $f$ is not continuous at zero meaning (by Theorem 6.2.6) that $f_n$ does not converge to $f$ uniformly.
  \item Each $g_n$ is continuous at zero, and the pointwise limit $g$ is also continuous at zero. Since we aren't contradicting 6.2.6 the convergence may or may not be uniform.

The definitions show $|g(x)-g_n(x)| < 1/n$ for all $x$ (max is at $x=1/(n+1)$).
    Setting $N > 1/\epsilon$ gives (for all $n \ge N$ and for all $x \in \mathbf{R}$)
    $$
    |g(x)-g_n(x)| < \epsilon
    $$
    As desired, thus $(g_n) 	o g$ uniformly.
  \item Each $h_n$ is continuous at zero, and so is the pointwise limit $h$. 6.2.6 doesn't apply so we'll have to check if the convergence is uniform.
    Notice that if $x_n = 1/n$ then
    $$
    |h(x_n) - h_n(x_n)| = 1 - 1/n
    $$
    For all $n$, meaning no matter how big $n$ is, we can't make $|h-h_n|<1/2$ for all $x$ implying $h_n$ \emph{does not} converge to $h$ uniformly.
   \end{enumerate}

Exercise 6.2.2

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Exercise 6.2.2

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