Exercise 4.4.13

Question

[Continuous Extension Theorem]
  \begin{enumerate}[label=(\alph*)] 
  \item Show that a uniformly continuous function preserves Cauchy sequences; that is, if $f: A \rightarrow \mathbf{R}$ is uniformly continuous and $\left(x_{n}\right) \subseteq A$ is a Cauchy sequence, then show $f\left(x_{n}\right)$ is a Cauchy sequence.
  \item Let $g$ be a continuous function on the open interval $(a, b)$. Prove that $g$ is uniformly continuous on $(a, b)$ if and only if it is possible to define values $g(a)$ and $g(b)$ at the endpoints so that the extended function $g$ is continuous on $[a, b]$. (In the forward direction, first produce candidates for $g(a)$ and $g(b)$, and then show the extended $g$ is continuous.)
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Let $\epsilon > 0$ and set $N$ large enough that $n,m \ge N$ has $|x_n - x_m| < \delta$ implying $|f(x_n) - f(x_m)| < \epsilon$ by the uniformly continuity of $f$.
  \item Define $g(a) = \lim_{x 	o a} g(x)$ and $g(b) = \lim_{x 	o b} g(x)$ if both limits exist, then $g$ is continuous on $[a,b]$ meaning it is uniformly continuous on $[a,b]$ by Theorem 4.4.7, and thus is uniformly continuous the subset $(a,b)$.

If $f$ were uniformly continuous $(a,b)$ Cauchy sequences are preserved meaning the sequential definition for functional limits (Theorem 4.2.3) implies the limits $\lim_{x 	o a} g(x)$ and $\lim_{x 	o b} g(x)$ exist.
   \end{enumerate}

Exercise 4.4.13

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Exercise 4.4.13

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